Geometry Study Guide
The distance formula can be used to prove a triangle has congruent sides a right angle parallel sides congruent angles
congruent sides
sine 37
cosine 53
inverse function: tangent
cotangent (tan -1)
Calculating the distance between two points
d= √(x2-x1)2+(y2-y1)2
Arc length
distance between 2 points on the edge of a circle
Pythagorean Theorem
finding the missing side
Square Perimeter
the 4 sides added together
circumference
the distance around the edge of a circle
slant height
the distance from the vertex to the base along the surface of the figure (cone/pyramid)
right cylinder
the height or altitude can be drawn so that it connects the centers of the circular bases
right cone
the height or altitude can be drawn so that it connects the tip of the cone with the center of the base
circumference formula for a circle
c= diameter times pi
Find the point that splits segment EF in half if point E is located at (5, −2) and point F is located at(−1, 3). (1.5, 1) (2, 0.5) (3.5, 4.5) (3, −2.5)
(2, 0.5)
A circle has its center at (1, 4) and a radius of 2 units. What is the equation of the circle? (x − 1)2 + (y − 4)2 = 4 (x + 2)2 + (y + 4)2 = 2 (x + 1)2 + (y − 4)2 = 4 (x − 1)2 + (y − 4)2 = 2
(x − 1)2 + (y − 4)2 = 4
Point G is located at (3, −1), and point H is located at (−2, 3). Find the point that is the distance from point G to point H. (0.33, −1.67) (−0.33, 1.67) (−0.5, −1) (6.33, −2.67)
(−0.33, 1.67)
Find the point C such that AC and BC form a 2:3 ratio. (−1, 1.2) (−0.6, 3) (0, 2.4) (0.5, 2)
(−0.6, 3)
Line AB contains points A (−2, 6) and B (4, 5). The slope of line AB is −6 6 -1/6 1/6
-1/6
If the sin 90° = 1, then the cos 0° = _____. 0, because the angles are complementary 1/2, because the angles are complementary square root of 3/2, because the angles are complementary 1, because the angles are complementary
1, because the angles are complementary
Box A has a volume of 12 cubic meters. Box B is similar to box A. To create Box B, Box A's dimensions were multiplied by five. What is the volume of Box B? 60 m3 300 m3 1,500 m3 7,500 m3
1,500 m3
Concentric Circles
circles with the same center
triangle area
1/2 times base times height
Volume of a pryamid
1/3 times base times height
Volume of a cone
1/3 times pi times radius squared times height
Find the perimeter of the following shape: 10.8 11 11.4 11.6
10.8
The length of the shadow of a building is 100 meters, as shown below: What is the height of the building? (1 point) 50 m 100 m 200 m 55 m
100 m
A source of laser light sends rays AB and AC toward two opposite walls of a hall. The light rays strike the walls at points B and C, as shown below: What is the distance between the walls? 110.90 m 149.28 m 103.92 m 173.20 m
103.92 m
Segment RS is an altitude of triangle PQR. Find the area of the triangle. 15.5 16 17.5 18
16
A quadrilateral PQRS is inscribed in a circle, as shown below: What is the measure of arc PQR? 110° 140° 70° 220°
220°
The density of a fish tank is 0.4 . There are 12 fish in the tank. What is the volume of the tank? 3 ft3 30 ft3 48 ft3 96 ft3
30 ft3
The picture shows a barn door: What is the distance BC between the two horizontal parallel bars? 4/ cos 45° 4 sin 45° 4/ tan 45° 4 tan 45°
4 tan 45°
Two birds sit at the top of two different trees 57.4 feet away from one another. The distance between the second bird and a bird watcher on the ground is 49.6 feet. What is the angle measure, or angle of depression, between the first bird and the bird watcher? 30.2° 59.8° 40.8° 49.2°
40.8°
The dimensions of a conical funnel are shown below: Nisha closes the nozzle of the funnel and fills it completely with a liquid. She then opens the nozzle. If the liquid drips at the rate of 15 cubic inches per minute, how long will it take for all the liquid to pass through the nozzle? (Use π = 3.14.) 3.14 minutes 5.02 minutes 2.51 minutes 3.76 minutes
5.02 minutes
Two ropes, AD and BD, are tied to a peg on the ground at point D. The other ends of the ropes are tied to points A and B on a flagpole, as shown below: Angle ADC measures 60° and angle BDC measures 45°. What is the distance between the points A and B on the flagpole? 6.59 feet 1.73 feet 4.32 feet 2.56 feet
6.59 feet
What is the arc length of in the circle below? 3.14 feet 4.88 feet 6.98 feet 11.78 feet
6.98 feet
The circle shown below has AB and BC as its tangents: If the measure of arc AC is 120°, what is the measure of angle ABC? 60° 70° 40° 65°
60°
A cone has a volume of 24 cubic inches. What is the volume of a cylinder that the cone fits exactly inside of? 8 in3 24 in3 48 in3 72 in3
72 in3
Find the diameter of a cone that has a volume of 83.74 cubic inches and a height of 5 inches. Use 3.14 for pi. 3 inches 4 inches 8 inches 16 inches
8 inches
The picture below shows a container that Rene uses to freeze water: What is the minimum number of identical containers that Rene would need to make 2000 cm3 of ice? (Use π = 3.14.) 27 9 14 20
9
The volume of a fish tank is 30 cubic feet. If the density is 0.3 , how many fish are in the tank? 100 10 9 3
9
How do you construct the inscribed and circumscribed circles of a triangle and what do you know about opposite angles of inscribed quadrilaterals?
A circumscribed circle is a circle that surrounds a polygon and intersects each one of its vertices. You can construct a circumscribed circle using a compass and straightedge by drawing perpendicular bisectors to the sides of the polygon. The point of intersection of these perpendicular bisectors is also the center of the circumscribed circle, known as the circumcenter. From the circumcenter, the circle is constructed with the compass set equal to the distance to any of the polygon's vertices. An inscribed circle is a circle that is contained within the interior of a polygon and intersects each side of a polygon exactly one time at a 90° angle. Therefore, the sides of the polygon represent tangents to the circle. While an inscribed circle may be constructed in any triangle using its angle bisectors, it can only be constructed within regular polygons with more than three sides. In these instances, the intersection of the perpendicular bisectors of the regular polygon marks the incenter, or center, of the inscribed circle. From the incenter, the circle is constructed with the compass set equal to the distance to any of the intersection points between a polygon side and its perpendicular bisector. When a quadrilateral is inscribed in a circle, the opposite angles are supplementary.
distance
A formula that can be used to find the length between two points
midpoint
A point that lies equal distance between two other points
base of a cylinder
b= pi times radius squared
parallelogram area
base x height
Amy and Fraser walk inside a circular lawn. Point O is the center of the lawn, as shown below: Amy walks from point B to point C along the segment BC. Fraser walks from point O to point A along the segment OA. Which statement is correct? Both Amy and Fraser walk distances equal to the diameter of the lawn. Amy walks a distance equal to the radius, and Fraser walks a distance equal to the diameter of the lawn. Both Amy and Fraser walk distances equal to the radius of the lawn. Amy walks a distance equal to the diameter, and Fraser walks a distance equal to the radius of the lawn.
Amy walks a distance equal to the diameter, and Fraser walks a distance equal to the radius of the lawn.
If a rectangle is rotated about the y-axis as shown below, what three-dimensional shape would be formed? (1 point) Rectangular prism Cone Cylinder Square pyramid
Cylinder
equilateral triangle
Equilateral triangles have three congruent sides. Equi- sounds like "equal" for a reason. They mean the same thing! And lateral means sides. So this literally means "equal sides."
How do you write an equation of a line so that it is parallel or perpendicular to a given point and a given line?
First, you need to determine the slope of the given line. Then, you can use the point-slope formula to write and equation of a line parallel by making the slopes the same and plugging in x and y to the formula. You can make an equation of a line perpendicular by making the slopes opposites and reciprocals and plugging in x and y to the formula.
depression
Horizontal line going down
elevation
Horizontal line going up
Exterior Angle to a Circle Theorem
If two secants, two tangents, or a secant and a tangent intersect outside a circle, the measure of the created angle between them is one-half the absolute value of the difference of the measures of their intercepted arcs
What is the relationship among inscribed angles, radii, and chords? Central, circumscribed and inscribed angles? inscribed angles and diameter? Radii and tangents?
Inscribed angles are ½ of the sum of the two intersecting arcs. Radii are perpendicular to the tangent where it intersects the circle. Inscribed angles form right angles if they are on the diameter. A circumscribed angle is 180- the central angle. Inscribed angles are ½ of the central angle.
isosceles triangle
Isosceles triangles have at least two congruent sides. If you fold an isosceles triangle in half between the two sides that are the same, both halves of the triangle match up. This means that all isosceles triangles are symmetrical.
What does a two-dimensional cross-section of three-dimensional object look like and how is that related to Cavalieri's Principle?
It depends on the 3-D object you start with and what type of cross section you are taking: Cylinder Parallel cross section = circle Perpendicular cross section = rectangle Diagonal cross section = ellipse Cone Parallel cross section = circle Perpendicular cross section (lying on the diagonal) = triangle Diagonal cross section = ellipse Sphere Parallel cross section = circle Perpendicular cross section = circle Diagonal cross section = circle Square Pyramid Parallel cross section = square Perpendicular cross section = triangle Diagonal cross section = trapezoid The cross sections relate to Cavalieri's principle because if the area of the cross sections of two 3-D figures are congruent and the height of the figures is also congruent, then it can be concluded that the volumes of the two figures are congruent no matter how they are presented.
Which statement best describes a radius of a circle? It is a segment that connects two distinct points on the secant of the circle. It is a line that intersects the circle at exactly two points on the diameter. It is a line that touches two points on the circle. It is a segment from a point on the circle to the center of the circle.
It is a segment from a point on the circle to the center of the circle.
rectangle area
Length x Width
Rectangle Perimeter
P=2l+2w
cross section of a cone
Parallel cross section = circle Perpendicular cross section (lying on the diameter) = triangle Diagonal cross section = ellipse
cross section of a sphere
Parallel cross section = circle Perpendicular cross section = circle Diagonal cross section = circle
cross section of a cylinder
Parallel cross section = circle Perpendicular cross section = rectangle Diagonal cross section = ellipse
cross section of a square pyramid
Parallel cross section = square Perpendicular cross section = triangle Diagonal cross section = trapezoid
Quadrilateral ABCD has opposite sides that are parallel and side AB congruent to side DC. What classification can be given to ABCD? Parallelogram Rectangle Rhombus Square
Parallelogram
What step is similar when constructing a circle inscribed in a triangle and a circle circumscribed about a triangle? Construct the angle bisectors of each angle in the triangle. Construct the perpendicular bisectors of each side of the triangle. Place the compass on a vertex and use the bisectors to draw the circle. Place the compass on the intersection of the bisectors to draw the circle.
Place the compass on the intersection of the bisectors to draw the circle.
A circular plate has a crack along the line AB, as shown below: If A is the center of the plate, what is the segment AB called? Tangent of the plate Chord of the plate Radius of the plate Secant of the plate
Radius of the plate
right triangle
Right triangles have one 90°angle and two acute angles.
How do similar right triangles lead to the definitions of the trigonometric ratios?
Similar right triangles have sides that are proportional. If a small triangle had a proportion of (left side)/(right side) of 3/6, then a larger (similar) triangle could have a proportion of 6/12. The fraction 6/12 can be reduced to 3/6. We can then relate the sides to an angle and label them as opposite over hypotenuse, for example. This shows that similar triangles have similar trigonometric ratios.
When the coordinates (1, 1), (4, 4), (7, 1), and (4, −2) are joined, which shape is formed? Parallelogram Rectangle Rhombus Square
Square
Cavalieri's Principle
The area of the cross sections of 2 3D figures are congruent and the heights of the figures are also congruent, then it can be concluded that the volumes of the two figures are congruent
How are the formulas for the volume of a cylinder, pyramid, and cone derived?
The cylinder formula is made up of the area of a circle formula multiplied by the height of the "stack" or cylinder. The formula for a pyramid has its roots in the volume of a cube formula. A cube can be split into 6 equal pyramids, so you can take the volume of a cube and then find 1/6 of it to find the volume of a pyramid which is 1/3 the volume of a half cube. The formula for the area of a cone is derived from the taking 1/3 the volume of a cone.
How do you use the distance formula and slope formula to classify a quadrilaterals and triangles?
The distance formula can help determine the length of each side of the quadrilateral. Usually, we are looking to see if all sides are equal or if the opposite sides are equal. The slope formula is useful when trying to determine if sides are parallel or perpendicular (same slope lines are parallel, negative reciprocals for perpendicular). For triangles, the distance formula can help determine the length of each side of the triangle. Usually, we are looking to see if all sides are equal or if at least two sides are equal. The slope formula is useful when trying to determine if sides are perpendicular to determine if the triangle is a right triangle.
Secant Interior Angle Theorem
The measure of a secant angle is half the sum of the arcs it and its vertical angles intercept
slope
The ratio of change in y values to change in x values; the measure of the steepness of a line
Use the image below to answer the following question. What relationship do the ratios of sin x° andcos y° share? The ratios are opposites ( -6/10 and 6/10 ). The ratios are reciprocals ( 6/10 and 10/6 ). The ratios are both negative ( -10/6 and -10/6 ). The ratios are both identical ( 6/10 and 6/10 ).
The ratios are both identical ( 6/10 and 6/10 ).
What is the relationship between the sine and cosine of complementary angles and why is this relationship true?
The sine and cosine of complementary angles are congruent. They are congruent because they reference the same side. For example in triangle ABC with sides a,b,c (a and b are legs and c is the hypotenuse) and angles A and B are complementary, the sine of A is a/c while the cosine of B is a/c as well.
What is the shape of the cross section taken perpendicular to the base of a cone? Circle Rectangle Square Triangle
Triangle
Congruent Arc Theorem
Two arcs are congruent if the central angles that intercept the are also congruent
perpendicular lines
Two lines that intersect at 90-degree angles
parellel lines
Two lines that lie within the same plane and never intersect
How do you use trigonometric ratios to solve for a missing side or angle of a right triangle?
Two sides of a right triangle are known, you can use the Pythagorean Theorem to solve for the missing side. If one side and one angle are known, you can use a trigonometric ratio to solve for the missing sides. If two sides are known, you can also use trigonometric ratios to solve for the missing angles by using the inverse trig function on the calculator.
Rotations of 2-D cylinder
When a rectangle is rotated about an axis, it creates a cylinder
Rotations of 2-D cone
When a right triangle is rotated about an axis, it creates a cone
Secant-Tangent Intersection Theorem
When a secant and a tangent intersect at a point of tangency, the angels created at the point of intersection are half the measurement of the arcs they intersect
Rotations of 2-D sphere
When a semi-circle is rotated about an axis, it creates a sphere
How do changes in dimensions affect the volume of common geometric solids?
When all three dimensions are changed by the same amount in a figure, then the volume is multiplied by that factor three times, once for each dimension. For example, if a cube had all of its dimensions doubled, then the volume of the new cube would be (2x2x2) 8 times bigger
How do you prove two circles are similar?
You can prove two circles are similar by using the radius from each circle and setting up a ratio. At any point on the circle, the radius will remain the same, so no matter where you "measure" the radius, the measurement is always constant. This means the ratio of the two radii is always constant, therefore, any two circles are similar.
How do you use coordinates to find the perimeter and area of polygons?
You can use the coordinates of a figure, along with the distance formula, to find the length of sides of the figure. For perimeter, you simply need to add all the side lengths together to calculate perimeter. Depending on the shape, the formula for area will differ.
net drawing
a 2-dimensional pattern of a 3-dimensional figure
cross section
a 3-dimensional figure is the intersection of a solid by a plane
diameter
a straight line drawn from edge to edge of a circle passing through the center
radius
a straight line drawn from the center of the circle to the edge of the circle
area formula for a circle
a= radius squared times pi
cos angle
adjacent/hypotenuse
chord
any straight line drawn between two points that lie on a circle
The picture shows a triangular island: Which expression shows the value of c? b/ tan 45° b/ cos 45° a cos 45° a sin 45°
b/ cos 45°
pieces of right triangles
hypotenuse/leg, opposite/side opposite, and adjacent/angle opposite
Minor arc
identified by 2 points and an arc measuring less than 180°
Major arc
identified by 3 points and an arc measuring more than 180° but less than 360°
net drawing of a sphere
is made up of a series of elongated, pointed ellipses
net drawing of a cone
is made up of a small circle and a larger quarter-circle
net drawing of a square pyramid
is made up of four equally sized triangles and a square
net drawing of a cylinder
is made up of two circles and a rectangle
Secant
line crossing 2 points of a circle
Tangent
line touching a circle only 1 time and always at 90 degrees from the center
perpendicular line equation
lines are special because the slopes of the lines are negative reciprocals of each other
Calculating the slope of a line from a graph or two points
m= (vertical) change in y values/ (horizontal) change in x values, or rise/ run, or m= y2-y1/x2-x1
Finding the midpoint of a line segment
m= (x1+x2/2, y1+y2/2)
Density formula
mass/volume
Line FG contains points F (3, 7) and G (−4, −5). Line HI contains points H (−1, 0) and I (4, 6). Lines FG and HI are parallel perpendicular neither
neither
tan angle
opposite/adjacent
sin angle
opposite/hypotenuse
coordinates
ratios and distance
inverse function: sine
reciprocal cosecant (sine -1)
inverse function: cosine
reciprocal secant (cos -1)
Similar Volume
sides cubed
Look at the figure: If tan x° = and cos x° = what is the value of sin x°? sin x° = 4b sin x° = b/a sin x° = 4a sin x° = a/b
sin x° = a/b
cosine 37
sine 53
Secant Angles
sum of arcs divided by two
oblique cylinder
the height or altitude cannot be drawn so that it connects the centers of the circular bases. Recall that the height is always represented by a perpendicular segment
oblique cone
the height or altitude cannot be drawn so that it connects the tip of the cone with the center of the base
parallel line equation
the lines stay parallel because the slopes of the lines are the same
Inscribed Angle Theorem
the measure of an inscribed angle is equal to half the measure of its intercepted arc
parallelogram
two pairs of parallel sides
rectangle
two pairs of parallel sides and congruent angles (all equal 90°)
rhombus
two pairs of parallel sides and congruent sides (all sides equal)
square
two pairs of parallel sides, congruent angles (all equal 90°) and congruent sides (all sides equal)
Volume of a sphere
v= 4/3 times pi times radius cubed (r3)
Volume of a cylinder
v= base times height
Which equation could be used to solve for the measure of angle P? y + x = 180 y + w= 180 y − (z + w) = 360 y + z + w = 360
y + w= 180
The equation of line AB is y = 5x + 1. Write an equation of a line parallel to line AB in slope-intercept form that contains point (4, 5). y = 5x − 15 y = −5x + 15 y = 1/5x + 21/5 y = 1/5x − 29/5
y = 5x − 15
slope-intercept form
y=mx+b
