Measurement and Geometry
Use the coordinate plane of the given image below to reflect the image across the y-axis (SEE CARD FOR ORIGINAL IMAGE)
*PICTURE ON CARD IS DIFFERENT, BUT THIS ONE IS SIMILAR* To reflect the image across the y-axis, replace each x-coordinate of the points that are the vertex of the triangle, x, with its negative, -x.
Define the six basic trigonometric ratios of right triangles
*practice by drawing a triangle and writing yourself* SEE CARD FOR MORE DETAIL
Give the equivalent of the following fluid measurements: 1 cup in fluid ounces 1 pint in cups and fluid ounces
1 cup = 8 fluid ounces 1 pint = 2 cups = 16 fluid ounces
Give the equivalent of the following metric measurements: 1 liter in milliliters and cubic centimeters 1 meter in millimeters and centimeters 1 gram in milligrams 1 kilogram in grams Give the meanings of the metric prefixes kilo-, centi-, and milli-
1 liter = 1,000 ml = 1,000 cubic cm 1 m = 1,000 mm = 100 cm 1 g = 1,000 mg 1 kg = 1,000 g Kilo means 1,000 base units Centi means 1/100 base units. Milli means 1/1,000 base units
Give the equivalents for the following English measurements: 1 yard in feet and inches 1 mile in feet and yards 1 acre in square feet 1 quart in pints and cups 1 gallon in quarts, pints, and cups 1 pound in ounces 1 ton in pounds
1 yd = 3ft = 36 in 1 mile = 5280 ft = 1760 yd 1 acre = 43,560 square ft 1 quart = 2 pints = 4 cups 1 gallon = 4 quarts = 8 pints = 16 cups 1 lb = 16 ounces 1 ton = 2000 lb
Give the general name for each shape that contains from three sides to ten sides, inclusive, as well as a twelve-sided figure. Explain how to find the sum of the interior angles of a polygon, and the measure of one interior angle of a regular polygon.
3: triangle 4: quadrilateral 5: pentagon 6: hexagon 7: heptagon 8: octagon 9: nonagon 10: decagon 12: dodecagon n sides: n-gon To find the sum of the interior angles of a polygon, use the formula: sum of interior angles = (n - 2)*180 degrees, where n is the number of sides in the polygon. This formula works with all polygons, not just regular polygons. To find the measure of one interior angle of a regular polygon, use the formula (n - 2)*180 / n, where n is the number of sides in the polygon
Explain the terms inscribed and circumscribed as they relate to circles
A circle is inscribed in a polygon is each of the sides of the polygon is tangent to the circle. A polygon is inscribed in a circle is each of the vertices of the polygon lies on the circle A circle is circumscribed about a polygon is each of the vertices of the polygon lies on the circle. A polygon is circumscribed about the circle if each of the sides of the polygon is tangent to the circle. If one figure is inscribed in another, then the other figure is circumscribed about the first figure
Explain the following terms as the relate to polygons: diagonal, convex, concave, and tell how to determine the number of diagonals a polygon has
A diagonal is a line segment that joins two non-adjacent vertices of a polygon A convex polygon is a polygon whose diagonals all lie within the interior of the polygon, A concave polygon is a polygon with at least one diagonal that lies outside the polygon. The number of diagonals a polygon has can be found by using the formula n*(n-3) / 2, where n is the number of sides in the polygon. This formula works for all polygons, not just regular polygons
Define dilation of a figure
A dilation is a transformation which proportionally stretches or shrinks a figure by a scale factor. The dilated image is the same shape and orientation as the original image but a different size. A polygon and its dilated image are similar
Give the geometric description of a horizontal hyperbola. Include the equation of a hyperbola, vertices, foci, center, and asymptotes
A hyperbola is the set of all points in a plane, whose distance from two fixed points, called foci, has a constant difference The standard equation of a horizontal hyperbola is ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1, where ,b, h, and k are real numbers. The center is the point (h, k) the vertices are the points (h + a, k) and (h - a, k), the points that every point on one of the parabolic curves is equidistant from and are found using the formula (h + c, k) and (h - c, k), where c^2 = a^2 + b^2. The asymptotes are two lines the graphs of the hyperbola approaches but never reaches, and are given by the equations y = (b/a)*(x - h) + k and y = -(b/a)*(x - h) + k
Discuss the properties of a plane
A plane is a 2-D flat surface defined by three non-collinear point. A plane extends an infinite distance in all directions in those two dimensions. It contains an infinite number of points, parallel lines and segments, intersecting lines and segments, as well as parallel or intersecting rays. A plane will never contain a 3-D figure or skew lines. A plane may intersect a circular conic surface, such as a cone, to form conic sections, such as the parabola, hyperbola, circle or ellipse. Two given planes will either be parallel or they will intersect to form a line
List the properties of quadrilaterals that will identify them as specifically a parallelogram, rhombus, square, or rectangle
A quadrilateral whose diagonals bisect each other is a parallelogram. A quadrilateral whose opposite sides are parallel (2 pairs of parallel sides) is a parallelogram. A quadrilateral whose diagonals are perpendicular bisectors of each other is a rhombus. A quadrilateral whose opposite sides (both pairs) are parallel and congruent is a rhombus. A parallelogram that has a right angle is a rectangle. (Consecutive angles of a parallelogram are supplementary. Therefore if there is one right angle in a parallelogram, there are four right angles in the parallelogram.) A rhombus with one right angle is a square. Because the rhombus is a special form of parallelogram, the rules about the angles of a parallelogram also apply to the rhombus.
Explain the terms ray, angle, and vertex
A ray is a portion of a line extending from a point in one direction. It has a definite beginning, but no end. An angle is formed when two rays meet at a common point. It may be a common starting point, or it may be the intersection of rays, lines, and/or segments A vertex is the point at which two segments or rays meet to form an angle. If the angle is formed by intersecting rays, lines and/or line segments, the vertex is the point at which four angles are formed.
Describe reflection over a line and reflection over a point
A reflection of a figure over a line (a "flip") creates a congruent image that is the same distance from the line as the original figure but on the opposite side. The line of reflection is the perpendicular bisector of any line segment drawn from a point on the original figure to its reflected image (unless the point and its reflected image happen to be the same point, which happens when a figure is reflected over one of its own sides). A reflection of a figure in a point is the same as the rotation of the figure 180 degrees about that point. The image of the figure is congruent to the original figure. The point of reflection is the midpoint of a line segment which connects a point in the figure to its image (unless the point and its reflected image happen to be the same point, which happens when a figure is reflected in one of its own points)
Explain the following terms as they apply to right triangles: leg, hypotenuse, Pythagorean Theorem
A right triangle has exactly on right angle. (If a figure has more than one right angle, it must have more than three sides, since the sum of the three angles of a triangle must equal 180 degrees) The side opposite the right angle is called the hypotenuse. The other two sides are called the legs. The Pythagorean Theorem states a unique relationship among the legs and hypotenuse of a right triangle: a^2 + b^2 = c^2, where a and b are the lengths of the hypotenuse. Note that this formula will only work with right triangles. Do not attempt to use it with triangles that are not right triangles
Discuss angles formed by two secants
A secant is a line that intersects a curve in two points. Two secants may intersect inside the circle, on the circle, or outside the circle. When the two secants intersect on the circle, an inscribed angle is formed. When two secants intersect inside a circle, the measure of each of two vertical angles is equal to half the sum of the two intersected arcs. When two secants intersect outside of a circle, the measure of the angle formed is equal to half the difference of the two arcs that lie between the two secants.
Give the geometric description of a vertical hyperbola. Include the equation of a hyperbola, vertices, foci, center, and asymptotes
A vertical hyperbola is formed when a plane makes a vertical cut through two cones that are stacked vertex-to-vertex. The standard equation of a vertical hyperbola is ((y - k)^2) / a^2) - ((x - h)^2 / b^2) = 1, where a, b, k, and h are real numbers. The center is the point (h, k), the vertices are the points (h, k+a) and (h, k-a), and the foci are the points that every point on one of the parabolic curves is equidistant from and are found using the formulas (h, k + c) and (h, k - c), where approaches but never reach, and are given by the equations y = (a/b)*(x - h) + k and y = -(a/b)*(x - h) + k
Explain the difference between alternate interior angles and alternate exterior angles as they relate to parallel lines with a transversal
Alternate Interior Angles: When two parallel lines are cut by a transversal, two interior angles that are on opposite sides of the transversal and on opposite parallel lines are congruent opposite interior angles. See angles3 & 6, 4 & 5. Alternate interior angles formed by parallel lines are congruent Alternate Exterior Angles: When two parallel lines are cut by a transversal, two exterior angles that are on opposite sides of the transversal and on opposite parallel lines are congruent opposite exterior angles. See angles 1 & 8, and 2 & 7. Alternate exterior angles formed by parallel lines are congruent
Define the terms altitude, height, concurrent, and orthocenter as they relate to triangle
Altitude of Triangle: A line segment drawn from one vertex perpendicular to the opposite side. In the diagram, BE, AD, and CF are altitudes Height of a Triangle: The length of the altitude, although the two terms are often used interchangeably Concurrent: Lines that intersect at one point. In a triangle, the three altitudes are concurrent Orthocenter of a Triangle: The point of concurrency of the altitudes of a triangle. Note that in an obtuse triangle, the orthocenter is the vertex of the right angle
Give the geometric description of an ellipse that is taller than it is wide. Include the equation of the ellipse, its center, axes, foci, and eccentricity
An ellipse is the set of all points in a plane, whose total distance from two fixed points called the foci (singular: focus) is constant, and whose center is the midpoint between the foci *SEE CARD FOR MORE DETAIL* The major axis has a length of 2a, and the minor axis has a length of 2b. Eccentricity (e) is a measure of how elongated an ellipse is, and is the ratio of the distance between the foci to the length of the major axis. Eccentricity will have a value between 0 and 1. The closer to 1 the eccentricity is, the closer the ellipse is to being a circle. The formula for eccentricity is c=a
Use the coordinate plane to create a dilation of the given image below, where the dilation is the enlargement of the original image *USE FLASHCARD FOR IMAGE*
An enlargement can be found by multiplying each coordinate of the coordinate pairs located at the triangles vertices by a constant. If the figure is enlarged by a factor of 2, the new image would be: *see card*
Explain the relationship between inscribed angles and intercepted arcs in a circle. Describe the angles inscribed in a semicircle.
An inscribed angle is an angle whose vertex lies on a circle and whose legs contain chords of that circle. The portion of the circle intercepted by the legs of the angle is called the intercepted arc. The measure of the intercepted arc is exactly twice the measure of the inscribed angle. In the first picture, angle ABC is an inscribed angle. arc AC = 2(m* angle ABC) Any angle inscribed in a semicircle is a right angle. The intercepted arc is 180 degrees, making the inscribed angle half that, or 90 degrees. In the second picture, angle ABC is inscribed in semicircle ABC, making angle B equal to 90 degrees
Explain approximate error and maximum possible error
Approximate Error: The amount of error in a physical measurement; often reported as the measurement, followed by the ± symbol and the amount of the approximate error Maximum Possible Error: Half the magnitude of the smallest unit used in the measurement. For example, if the unit of measurement is 1 centimeter, the maximum possible error is 1/2 cm, written as ±0.5 cm following the measurement. It is important to apply significant figures on reporting maximum possible error. Do not make the answer appear more accurate than the least accurate of your measurements
Define the following terms as they relate to circles: center, radius, and diameter
Center: A single point that is equidistant from every point on a circle Radius: A line segment that joins the center of the circle and any one point on the circle. All radii of a circle are equal Diameter: A line segment that passes through the center of the circle and has both endpoints on the circle. The length f the diameter is exactly twice the length of the radius
Explain central angles, major arcs, and minor arcs. Tell how many degrees are in a semicircle.
Central Angle: An angle whose vertex is the center of a circle and whose legs intercept an arc of the circle. Major Arc: An arc of a circle, having a measure of at least 180 degrees. The measure of the major arc can be found by subtracting the measure of the central angle from 360 degrees Minor Arc: an arc of a circle, having a measure less than 180 degrees. The measure of the central angle is equal to the measure of the arc. Semicircle: An arc having a measure of exactly 180 degrees
Define chord, secant, tangent, and point of tangency as they relate to circles
Chord: A line segment that has both endpoints on a circle. Secant: A line that passes through a circle and contains a chord of that circle. Tangent: A line in the same plane as a circle that touches the circle at exactly one point. While a line segment can be tangent to a circle as part of a line that is tangent, it is improper to say a tangent can be a line segment by itself that touches the circle in exactly one point. Point of Tangency: The point at which a tangent touches a circle
Discuss complementary, supplementary, and adjacent angles
Complementary: Two angles whose sum is exactly 90 degrees. The two angles may or may not be adjacent. In a right triangle, the two acute angles are complementary Supplementary: Two angles whose sum is exactly 180 degrees. The two angles may or may not be adjacent. Two intersecting lines always form two pairs of supplementary angles. Adjacent supplementary angles will always form a straight line Adjacent: Two angles that have the same vertex and share a side. Vertical angles are not adjacent because they share a vertex but no common side
Describe concentric circles. Define arc and semicircle.
Concentric circles are circles that have the same center, but not the same length radii. A bulls-eye target is an example of concentric circles. an arc is a portion of a circle. Specifically, an arc is the set of pints between and including two points on a circle. An arc does not contain any points inside the circle. When a segment is drawn from the endpoints of an arc to the center of the circle, a sector is formed. A semicircle is an arc whose endpoints are the endpoints of the diameter of circle. A semicircle is exactly half of a circle.
Define the following terms as they relate to polygona: side, vertex, regular polygon, apothem, and radius
Each straight line segment of a polygon is called a side. The point at which two sides of a polygon intersect is called the vertex. In a polygon, the number of sides is always equal to the number of vertices A polygon with all sides congruent and all angles equal is called a regular polygon. A line segment from the center of a polygon perpendicular to a side of the polygon is called the apothem. In a regular polygon, the apothem can be used to find the area of the polygon using the formula A = 0/5*ap, where a is the apothem and p is the perimeter. A line segment from the center of the polygon to a vertex of the polygon is called the radius. The radius of a regular polygon is also the radius of a circle that can be circumscribed about the polygon.
Describe the following types of triangles: equilateral, isosceles, scalene
Equilateral Triangle: triangle with three congruent sides and angles Isosceles Triangle: triangle with two congruent sides and angles. The congruent angles are opposite the congruent sides Scalene Triangle: triangle with no congruent sides and no congruent angles. The angle with the largest measure is opposite the longest side; the angle with the smallest measure is opposite the smallest side
Explain interior angles, exterior angles, and corresponding angles on terms of a parallel lines with a transversal
Interior Angles: When two parallel lines are cut by a transversal, the angles that are between the 2 parallel lines are interior angles. In the diagram, angles 3, 4, 5, and 6 are interior angles Exterior Angles: When two parallel lines are cut by a transversal, the angles that are outside the parallel lines are exterior angles. In the diagram below, angles 1, 2, 7, and 8 are exterior angles Corresponding Angles: When 2 parallel lines are cut by a transversal, the angles that are in the same position relative to the transversal, the angles that are in the same position relative to the transversal, and one of the parallel lines. The diagram below has four pairs of corresponding angles: angles 1 and 5; angles 2 and 6; angles 3 and 7; and angles 4 and 8. Corresponding angles formed by parallel lines are congruent.
Explain the relationship between intersecting lines, parallel lines, vertical angles, and transversals
Intersecting Lines" Lines that have exactly one point in common Parallel lines: lines in the same plane that have no points in common and never meet. It's possible for lines to be in different planes, have no points in common, and never meet, but they are not parallel because they are in different planes Vertical angles: Non-adjacent angles formed when two lines intersect. Vertical angles are congruent. Transversal: a straight line that intersects at least two other lines, which may or may not be parallel
Discuss the terms symmetry, symmetric, and line of symmetry
Line of Symmetry: The line that divides a figure or object into two symmetric parts. Each symmetric half is congruent to the other. An object may have no lines of symmetry, one symmetry, or more than one line of symmetry
Explain median and centroid as they relate to triangles
Median of a Triangle: a line segment drawn from one vertex to the midpoint of the opposite side. This is not the same as the altitude, except the altitude to the base of an isosceles triangle and all three altitudes of an equilateral triangle Centroid of the Triangle: The point of concurrency of the medians of a triangle. This is the same point as the orthocenter only in an equilateral triangle. The centroid can also be considered the exact center of the triangle. Any shape triangle can be perfectly balanced on a tip placed at the centroid. The centroid is also the point that is two-thirds the distance from the vertex to the opposite side
Give the geometric description of a parabola. Include the equation of a parabola, the directrix, focus, vertex, and axis
Parabola: The set of all points in a plane that are equidistant from a fixed line, called the directrix, and fixed point not on the line, called the focus Axis: The line perpendicular to the directrix that passes through the focus For parabolas that open up or down, the standard equation is (x - h)^2 = 4cv(y - k), where h, c, and k are coefficients. If c is positive, the parabola opens up. If c is negative, the parabola opens down. The vertex is the point (h, k). The directrix is the line having the equation y = -c + k, and the focus is the point(h, c+k) For parabolas that open left or right, the standard equation is (y - k)^2 = 4c(x - h), where k, c, and h are coefficients. If c is positive, the parabolas opens to the right. If c is negative, the parabola opens to the left. The vertex is the point (h, k). The directrix is the line having the equation x = -c + h, and the focus is the point (c+h, k)
Discuss perpendicular bisectors and angle bisectors as they relate to triangles
Perpendicular Bisector: A line that bisects the side of a triangle at a right angle. The perpendicular bisectors of a triangle are concurrent at a point called the circumcenter that is equidistant from the three vertices. The circumcenter is also the center of the circle that can be circumscribed about the triangle Angle Bisector: A line divides the vertex angle of a triangle into two equal parts. The angle bisectors are concurrent at a point called the incenter that is equidistant from the three sides. The incenter is also the center of the largest circle that can be inscribed in the triangle
Describe perpendicular lines and perpendicular bisectors
Perpendicular lines are lines that intersect at right angles. They are represented by the symbol ⊥. The shortest distance from a line to a point not on the line is a perpendicular segment from the point to the line. In a plane, the perpendicular bisector of a line segment is a line comprised of the set of all points that are equidistant from the endpoints of the segment. This line always forms a right angle with the segment on the exact middle of the segment. Note that you can only find perpendicular bisectors of segments
Define the following geometric terms: point, line, plane, collinear, coplanar, ray, line segment, angle, transversal, perpendicular, parallel
Point: a fixed location in space; has no size or dimensions; commonly represented by a dot Line: a set of points that extends infinitely in two opposite directions; has length, but no width or depth; a line can be defined by two or more distinct points that it contains Plane: a two dimessional surface that extends infinitely in all available directions; a plane can be defined by any three distinct points that it contains, or any line and another point not on that line Collinear: multiple points that lie on the same line Coplanar: multiples points or lines that lie on the same plane Ray: a part of a line that has one endpoint and infinite length; defined by the single endpoint and a direction Line Segment: a part of a line that has two endpoints and a fixed length; defined by two endpoints Angle: formed by two intersecting lines/rays/segments; defines the difference in orientation between the two; most commonly measured in degrees Transversal: a line that crosses two or more other lines Perpendicular: a line or plane that intersects another line or plane at a right (90 degree) angle Parallel: a set of lines or planes that never intersect and are the same distance apart at every point
Explain the difference between precision and accuracy
Precision: How reliable and repeatable a measurement is. The more consistent the data is with repeated testing, the more precise it is. For example, hitting a target consistently in the same spot, which may or may not be the center of the target, is precise Accuracy: How close the data is to the correct data. For example, hitting a target consistently in the center of the target, whether or not it hits the same spot, is accuracy. Note that it's possible for data to be precise without being accurate. If a scale is off balance, the data will be precise but not accurate. For data to have precision and accuracy, it must be repeatable and correct
Discuss rectangles, rhombuses, and squares
Rectangles, rhombuses, and squares are all special forms of parallelograms Rectangle: a parallelogram with four right angles. All rectangles are parallelograms, but not all parallelograms are rectangles. The diagonals of a rectangle are congruent Rhombus: A parallelogram with four congruent sides. All rhombuses are parallelograms, but not all parallelograms are rhombuses. The diagonals of a rhombus are perpendicular to each other. Square: A parallelogram with four congruent sides. All squares are also parallelograms, rhombuses, and rectangles. The diagonals of a square are congruent and perpendicular to each other
List the four types of congruent triangles
SSS: Three sides of one triangle are congruent to the three corresponding sides of the second triangle. SAS: Two sides and the included angle (the angle formed by those two sides) of one triangle are congruent to the corresponding two sides and included angle of the second angle ASA: Two angles and the included side (the side that joins the two angles) of one triangle are congruent to the corresponding two angles and included side of the second triangle. AAS: Two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of the second triangle. Note that AAA is not a form for congruent triangles. This would say that the three angles are congruent, but says nothing about the sides. This meets the requirements for similar triangles, but not congruent triangles.
Explain the triangle inequality theorem
The Triangle Inequality Theorem states that the sum of the measures of any two sides of a triangle is always greater than the measure of the third side. If the sum of the measures of two sides were equal to the third side, a triangle would be impossible because the two sides would lie flat across the third side and there would be no vertex. If the sum of the measures of two of the sides was less than third side, a closed figure would be impossible because the two shortest sides would never meet.
Tell how to find the arc length when the angle measure is given in radians. Define sectors as it relates to circles.
The arc length of a circle is the length of a portion of the circumference between two points on the circle. When the arc is defined by two radii forming a central angle, the formula for arc is defined by two radii forming a central angle, the formula for arc length is s = rθ, where s is the arc length, r is the radius, and v is the measure of the central angle in radians A sector is the portion of a circle formed by two radii and their intercepted arc. While the arc length is exclusively the points that are also on the circumference of the circle, the sector is the entire area bounded by the arc and the two radii
Tell how to find the area and perimeter of a parallelogram
The area of a parallelogram is found by the formula A = bh, where A is the area, b is the length of the base, and h is the height. Note that the base and height correspond to the length and width in a rectangle, so this formula would apply to rectangles as well. The perimeter of a parallelogram is found by the formula P = 2a + 2b or P = 2(a + b), where P is the perimeter, and a and b are the lengths of the two sides Do not confuse the height of a parallelogram with the length of the second side. The two are only the same measure in the case of a rectangle
Tell how to find the are and arc length of a sector of a circle
The area of a sector of a circle is found by the formula, A = (θr^2)/2, where A is the area, θ is the measure of the central angle in radians, and r is the radius. To find the area when the central angle is in degrees, use the formula A = (θπr^2)/360, where θ is the measure of the central angle in degrees and r is the radius. The arc length of a sector of a circle is found by the formula: arc length = rθ, where r is the radius and θ is the measure of the central angle in radians. To find the arc length when the central angle is given in degrees, use the formula: arc length = (θ(2πr))/360, where θ is the measure of the central angle in degrees and r is the radius
Tell how to find the area and perimeter of a trapezoid
The area of a trapezoid is found by the formula A = 0.5*h(b_1 + b_2), where A is the area, h is the height (segment joining and perpendicular to the parallel bases), and b_1 and b_2 are the two parallel sides (bases). Do not use one of the other two sides as the height unless that side is also perpendicular to the parallel bases. The perimeter of a trapezoid is found by the formula P = a + b_1 + c + b_2, where P is the perimeter, and a, b_1, c, and b_2 are the four sides of the trapezoid. Notice that the height does not appear in this formula
Tell how to find the are and perimeter of a n equilateral triangle
The area of an equilateral triangle is found by the formula A = sqrt(3)/4 * s^2, where A is the area ands is the length of a side. You could use the 30-60-90 ratios to find the height of the triangle and then use the standard triangle area formula, but this is faster. The perimeter of an equilateral triangle is found by the formula P = 3s, where P is the perimeter and s is the length of a side If you know the length of the apothem (distance from the center of the triangle perpendicular to the base) and the length of a side, you can use the formula A =0.5ap, where a is the length of the apothem and p is the perimeter
Tell how to find the area and perimeter of an isosceles triangle
The area of an isosceles triangle is found by the formula A = 0.5b*sqrt(a^2 - (b^2 / 4)), where A is the area, b is the base (the unique side), and a is the length of the one of the two congruent sides, If you do not remember this formula, you can use the Pythagorean theorem to find the height so you can use the standard formula for the area of a triangle. The perimeter of an isosceles triangle is found by the formula P = 2a + b, where P is the perimeter, a is the length of one of the congruent sides, and b is the base (the unique side)
Tell how to find the lateral surface area and volume of a sphere
The lateral surface area is the area around the outside of the sphere. The lateral surface area is given by the formula A = 4πr^2, where r is the radius. The answer is generally given in terms of pi. A sphere does not have separate formulas for lateral surface area and total surface area as other solid figures do. Often, a problem may ask for the surface area of a sphere. Use the above formula for all problems involving the surface area of a sphere. The volume is given by the formula V = (4/3)*πr^3, where r is the radius
Discuss projection of a point on a line and projection of a segment on a line
The projection of a point on a line is the point at which a perpendicular line is drawn from the given point to the given line intersects the line. This is also the shortest distance from the given point to the line. The projection of a segment on a line is a segment whose endpoints are the endpoints of the given segment to the given line. This is similar to the length a diagonal line appears to be when viewed from above
Tell how to find the volume and total surface area of a rectangular prism
The volume of a rectangular prism is found by the formula V = lwh, where V is the volume, l is the length, w is the width, and h is the height. Total surface area is the area of the entire outside surface of the solid. The total surface area of a rectangular prism is found by the formula TA = 2lw + 2lh + 2wh or TA = 2(lw + lh + wh), where TA is the total surface area, l is the length, w is the width, and h is the height. If the problem asks for lateral surface area, find the total area of the sides, but not the bases. Uses the formula LA = 2lh + 2wh or LA = 2(lh + wh), where l is the length, w is the width, and h is the height
Tell how to find the volume, lateral surface area, and total surface area of a right circular cone
The volume of a right circular cone is found by the formula V = 1/3(πr^2h), where V is the volume, r is the radius, and h is the height. Notice this is the same as 1/3 the volume of a right circular cylinder. The lateral surface area of a right circular cone is found by the formula LA = πr*sqrt(r^2 + h^2) or LA = πrs, where LA is the lateral surface area, r is the radius, h is the height, and s is the slant height (distance from the vertex to the edge of the circular base). s = sqrt(r^2 + h^2) The total surface area of a right circular cone is the same as the lateral surface area plus the area of the circular base. The formula for the total surface area plus the area of the circular base. The formula for total surface is TA = πr*sqrt(r^2 + h^2) + πr^2 or TA = πrs + πr^2, where TA is the total surface area, r is the radius, h is the height, and s is the slant height
Tell how to find the volume and total surface are of a right prism
The volume of a right prism is found by the formula V = Bh, where V is the volume, B is the area of the base, and h is the height (perpendicular distance between the bases). The total surface area is the area of the entire outside surface of a solid. The total surface area of a right prism is found by the formula TA = 2B + (sum of the areas of the sides), where TA is the total surface area and B is the area of one base. To use this formula, you must remember the formulas for the planar figures. If the problem asks for the lateral surface area (the area around the sides, not including the bases), use the formula LA = sum of the areas of the sides. Again, you will need to remember the formulas for the various planar figures.
Describe various evidence for transformations of figures
To identify that a figure has been rotated look for evidence that the figure is still face-up, but has changed it orientation To identify that a figure has been reflected across a line, look for evidence that the figure is now face-down To identify that a figure has been translated, look for evidence that a figure is still face-up and has not changed orientation; the only change is location. To identify that a figure has been dilated, look for evidence that the figure has changed its size but not its orientation
Explain transforming a given figure using rotation, reflection, and translation
To rotate a given figure: 1. identify the point of rotation. 2. Using tracing paper, geometry software, or by approximation, recreate the figure at a new location around the point of rotatino To reflect a given figure: 1. Identify the line of reflection 2. By folding the paper, using geometry software, or by approximation, recreate the image at a new location on the other side of the line of the reflection To translate a given figure: 1. Identify the new location. 2. Using graph paper, geometry software, or by approximation, recreate the figure in the the new location. If using graph paper, make a chart of the x- and y-values to keep track of the coordinates of all critical points
Tell how to find the volume and total surface area of a cube
V = s^3 V = volume, s = the length of the side. This is the same as the formula for the volume of a rectangular prism, except the length, width, and height are all equal TA = 6s^2 where TA = total surface area and s = length of a side
Tell how to find the volume, lateral surface area, and total surface area of a right circular cylinder
V = πr^2*h, where V is the volume, r is the radius, and h is the height. The lateral surface area is the surface area without the bases. The formula is LA = 2πrh, where LA is the lateral surface area, r is the radius, and h is the height. Remember that if you unroll a cylinder, the piece around the middle is a rectangle. The length of a side of the rectangle is equal to the circumference of the circular base, or 2πr. Substitute this formula for the length, and substitute the height of the cylinder for the width in the formula for the area of a rectangle. The total surface area of a cylinder is the lateral surface area plus the area of the two bases. The bases of a cylinder are circle, making the formula for the total surface area of a right circular cylinder TA = 2πrh + 2πr^2, where TA is the total area, r is the radius, and h is the height
Discuss quadrilaterals, parallelograms, and trapezoids
Video Code: 129981 Quadrilateral: A closed two-dimensional geometric figure comprised of exactly four straight sides. The sum of the interior angles of any quadrilateral is 360 degrees Parallelogram: A quadrilateral that has exactly two pairs of opposite interior angles are always congruent, and the consecutive interior angles are always congruent, and the consecutive interior angles are supplementary. The diagonals of a parallelogram into two congruent triangles Trapezoid: Traditionally, a quadrilateral that has exactly one pair of parallel sides. Some math texts define trapezoid as a quadrilateral that has at least one pair of parallel sides. Because there are no rules governing the second pair of sides, there are no rules that apply to the properties of the diagonals of a trapezoid.
Identify and describe the six types of angles based on angle measurement
Video Code: 204098 Acute: angle with a degree of less than 90 Right Angle: An angle with a degree of 90 Obtuse Angle: An angle with a degree greater than 90 but less than 180 Straight Angle: An angle with a degree of 180. aka semicircle Reflex Angle: An angle with a degree measure greater than 180 but less than 360 Full Angle: An angle with a degree of 360. aka full circle
Tell how to find the are, circumference, and diameter of a circle
Video Code: 2433015 The area of a circle is found by the formula A = πr^2, where A is the area and r is the length of the radius. If the diameter of the circle is given, remember to divide it in half to get the length of the radius before proceeding. The circumference of a circle is found by the formula C = 2πr, where C is the circumference and r is the radius. Again, remember to convert the diameter if you are given that measure rather than the radius. To find the diameter when you are given the radius, double the length of the radius
Explain the Law of Sines and the law of cosines
Video Code: 361120 The Law of Sines states that (sin A)/a = (sin B)/b = (sin C)/c, where A, B, and C are the angles of the triangle, and a, b, and c are the sides opposite their respective angles. This formula will work with ALL triangles. The Law of Cosines is given by the formula c^2 = a^2 + b^2 - 2ab(cos C) where a, b, and c are the sides of a triangle and C is the angle opposite of the side c. Can be used on ANY triangle.
Explain the relationship between the sides and angles of a triangle. Explain the relationship between the midpoints of the sides and the sides of a triangle. Describe similar triangles
Video Code: 398538 In any triangle, the angles opposite congruent sides are congruent, and the sides opposite congruent angles are congruent. The largest angle is always opposite the longest side, and the smallest angle is always opposite the smallest side. The line segment that joins the midpoints of any two sides of a triangle is always parallel to the third side and exactly half the length of the third side. Similar triangles are triangles whose corresponding angles are equal and whose corresponding sides are proportional. Represented by AA. Similar triangles whose corresponding sides are congruent are also congruent triangles.
Describe the following types of triangles: acute, right, and obtuse. Tell the sum of the angles of a triangle
Video Code: 511711 An acute triangle is a triangle whose three angles are all less than 90 degrees. If 2+ angles are equal, the acute triangle is also an isosceles triangle. If the three angles are all equal, the acute triangle is also an equilateral triangle A right triangle is a triangle with exactly one angle equal to 90 degrees. All right triangles follow the Pythagorean Theorem. A right triangle can never be acute or obtuse An obtuse triangle is a triangle with exactly one angle greater than 90 degrees. The other two angles may or may not be equal. If the two remaining angles are equal, the obtuse triangle is also an isosceles triangle. The sum of the measures of the interior angles of a triangle is always 180 degrees. Therefore, a triangle can never have more than one angle greater than or equal to 90 degrees
Describe rotation, center of rotation, and angle of rotation
Video Code: 602600 A rotation is a transformation that turns a figure around a point called the center of rotation, which can lie anywhere in the plane. If a line is drawn from a point on a figure to the center of rotation, and another line is drawn from the center to the rotated image of that point, the angle between the two lines is the angle of rotation. The vertex of the angle of rotation is the center of rotation.
Tell how to find the area and perimeter of a square
Video Code: 620902 The area of a square is found by using the formula A = s^2, where A is the area of the square, and s is the length of one side. The perimeter of a square is found by using the formula P = 4s, where P is the perimeter of the square, and s is the length of one side. Because all four sides are equal in a square, it is faster to multiply the length of the one side by 4 than to add the same number four times. You could use the formulas for rectangles and get the same answer.
Tell how to find the volume of a pyramid
Video Code: 621932 The volume of a pyramid is found by the formula V = 1/3*Bh, where V is the volume, B is the area of the base, and h is the height (segment from the vertex perpendicular to the base). Notice this formula is the same as 1/3 the volume of a right prism. In this formula, B represents the area of the base, not the length or width of the base. The base can be different shapes, so you must remember the various formulas for the areas of plane figures. In determining the height of the pyramid, use the perpendicular distance from the vertex to the base, not the slant height of one of the sides.
Explain the difference between congruent and similar figures
Video Code: 686174 Congruent figures are geometric figures that have the same size and shape. All corresponding angles are equal, and all corresponding sides are equal. It is indicated by the symbol ≅ Similar figures are geometric figures that have the same shape but do not necessarily have the same size. all corresponding angles are equal and all corresponding sides are proportional, but they do not have to be equal. It is indicated by the symbol ~.
Describe translation of a figure
Video Code: 718628 A translation is a transformation which slides a figure from one position in the plane to another position in the plane. The original figure and the translated figure have the same size, shape, and orientation
Tell how to find the area and perimeter of a triangle
Video Code: 853779 The area of a triangle is given by the formula A = 0.5*bh, where A is the area of the triangle, b is the length of the base, and h is the height of the triangle perpendicular to the base. If you know the three sides of a scalene triangle, you can use the formula. If you know three sides of a scalene triangle, you can use the formula A = sqrt(s*(s-a)*(s-b)*(s-c)), where A is the area, s is the semi perimeter s = (a + b + c)/2, and a, b, and c are the lengths of the three sides. In those cases, the triangle may be any shape. The variables a, b, and c aren't exclusive to right triangles in the perimeter formula
Tell how to find the area and perimeter of a rectangle
Video Code: 933707 The area of a rectangle is found by the formula A = lw, where A is the area of the rectangle, l is the length (usually considered to be the longer side) and w is the width (usually considered the shorter side). The numbers for l and w are interchangeable. The perimeter of a rectangle is found by the formula P = 2l +2w or p = 2(l + w), where P is the perimeter of the rectangle, l is the length, and w is the width. It may be easier to add the length and width first and then double the result, as in the second formula
