Geometry Semester 2 Review

Circumference

The distance around the circle. Like perimeter for a polygon

Resultant vector

The sum of two vectors is a resultant.

How can you prove two triangles are similar? (theorems/postulates)

1.) Angle-Angle Similarity (AA~) Postulate 2.) Side-Angle-Side Similarity (SAS~) Theorem 3.) Side-Side-Side Similarity (SSS~) Theorem

How do you draw a locus?

1.) Sketch several points that satisfy the stated condition(s) 2.) Keep doing so until you see a pattern 3.) Draw the figure the pattern suggests

What are some common Pythagorean triples?

A *Pythagorean triple* is a set of three nonzero whole numbers a, b, and c, that satisfy the equation a^2 + b^2 = c^2. 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

Transformation

A change in the position, size, or shape of a geometric figure. A transformation maps a figure onto its image. Prime notation is sometimes used to identify image points. (Ex. X', "X prime", is the image of X)

A composition of reflections across two intersecting lines is a ____.

A composition of reflections across two intersecting lines is a ROTATION.

A composition of reflections across two parallel lines is a ____.

A composition of reflections across two parallel lines is a TRANSLATION.

Composition of transformations

A composition of two transformations is a transformation in which a second transformation is performed on the image of a first transformation.

Symmetry

A figure has symmetry if there is an isometry that maps the figure onto itself.

Hemispheres

A great circle divides a sphere into two equal halves, hemispheres.

Secant

A line, ray, or segment that intersects a circle at two points.

Tangent to a circle

A line, segment, or ray in the plane of the circle that intersects the circle in exactly one point. That point is the point of tangency.

Volume

A measure of the space a figure occupies.

Pyramid

A polyhedron in which one face, the *base*, is a polygon and the other faces, the *lateral faces* are triangles with a common vertex, called the *vertex* of the pyramid. An *altitude* of a pyramid is the perpendicular segment from the vertex to the plane of the base. Its length is the *height* of the pyramid. A *regular pyramid* is a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles.

Prism

A polyhedron with two congruent and parallel faces, which are called the *bases*. The other faces, which are parallelograms, are called the *lateral faces*. An *altitude* of a prism is a perpendicular segment that joins the planes of the bases. Its length is the *height* of the prism. A *right prism* is one whose lateral faces are rectangular regions and a lateral edge is an altitude. In an *oblique prism*, some or all go the lateral faces are nonrectangular.

Geometric probability

A probability that uses a geometric model in which points represent outcomes.

Radius of a circle

A radius of a circle is any segment with one endpoint on the circle and the other endpoint at the center of the circle. It can also refer to the length of this segment.

Reflection

A reflection (flip) across line r is a transformation such that if point A is on line r, then the image of A is itself, and if a point B is not on line r, then its image B' is the point such that r is the perpendicular bisector of -BB'-

Tessellation

A repeating pattern of figures that completely covers a plane without gaps or overlap. Also called a *tiling* A pure tessellation is a tessellation that consists of congruent copies of one figure.

Rotation

A rotation (turn) of x degrees about a point R is a transformation such that for any point V, its image is the point V', where RV = RV' and m<VRV' = x. The image of R is itself.

Diameter of a circle

A segment that contains the center of the circle and whose endpoints are on the circle. It can also refer to the length of this segment.

Chord

A segment whose endpoints are on the circle.

Locus

A set of points, all of which meet a stated condition. Ex. The locus in red pictured: In a plane, the points 1 cm from segment -PQ-. (Two segments parallel to -PQ- and two semicircles centered at P and Q)

Proportion

A statement that two or more ratios are equal. An *extended proportion* is a statement that three or more ratios are equal.

Cone

A three-dimensional figure that has a circular *base*, a *vertex* not in the plane of the circle, and a curved lateral surface. The *altitude* of a cone is the perpendicular segment from the vertex to the plane of the base. The *height* is the length of the altitude. In a *right cone*, the altitude contains the center of the base.

Polyhedron

A three-dimensional figure whose surfaces, or *faces*, are polygons. The *vertices* of the polygons are the vertices of the polyhedron. The intersections of the faces are the *edges* of the polyhedron.

Cylinder

A three-dimensional figure with two congruent circular *bases* that lie in parallel planes. An *altitude* of a cylinder is a perpendicular segment that joins the planes of the bases. It length is the *height* of the cylinder. In a *right cylinder*, the segment joining the centers of the bases is an altitude. In an *oblique cylinder*, the segment joining the centers of the bases is not perpendicular to the planes containing the bases.

Isometry

A transformation in which an original figure and its image are congruent Also known as a *congruence transformation*

Dilation

A transformation that has center C and scale factor n, where n>0, and maps a point R to T' in such a way that R' is on -CR-> and CR' = n x CR. The center of dilation is its own image. Also called *similarity transformation* If n>1, the dilation is an enlargement, and if 0<n<1, the dilation is a reduction.

Translation

A transformation that moves points the same distance and in the same direction. (slide) A translation in the coordinate plane is described by a vector.

A translation or rotation is a composition of two ____.

A translation or rotation is a composition of two REFLECTIONS.

Adjacent arcs

Adjacent arcs are on the same circle and have exactly one point in common.

An angle inscribed in a semicircle is a ____ angle.

An angle inscribed in a semicircle is a RIGHT angle.

Inscribed angle

An angle is inscribed in a circle if the vertex of the angle is on the circle and the sides of the angle are chords of the circle.

Central angle of a circle

An angle whose vertex is the center of the circle.

Intercepted arc

An arc of a circle having endpoints on the sides of an inscribed angle, and its other points in the interior of the angle.

Major arc

An arc that is larger than a semicircle

Minor arc

An arc that is smaller than a semicircle

Vector

Any quantity that has *magnitude* (size) and direction. You can represent a vector as an arrow that starts at the *initial point* and points to the *terminal point*. A vector can be described by ordered pair notation {x, y}, where x represents horizontal change from the initial point to the terminal point and y represents vertical change from the initial point to the terminal point.

Congruent arcs

Arcs that have the same measure and are in the same circle or congruent circles.

Congruent circles

Circles whose radii are congruent.

Concentric circles

Concentric circles lie in the same plane and have the same center.

____ quadrilateral tessellates.

EVERY quadrilateral tessellates.

____ triangle tessellates.

EVERY triangle tessellates.

Adding vectors

For vector a = {x1, y1} and vector c = {x2, y2}, a+c = {x1+x2, y1+y2}.

Semicircle

Half a circle

If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is ____ to the circle.

If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is TANGENT to the circle.

Side-Splitter Theorem

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

If a line is tangent to a circle, then the line is ____ to the radius drawn to the point of tangency.

If a line is tangent to a circle, then the line is PERPENDICULAR to the radius drawn to the point of tangency.

Triangle-Angle-Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Side-Angle-Side Similarity (SAS~) Theorem

If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar.

Side-Side-Side Similarity (SSS~) Theorem

If the corresponding sides of two triangles are proportional, then the triangles are similar.

Perimeters and areas of similar figures

If the similarity ratio of two similar figures is a/b, then 1.) The ration of their perimeters is a/b 2.) The ratio of their areas is a^2/b^2

Areas and Volumes of Similar Solids

If the similarity ratio of two similar solids is a : b, then 1.) the ratio of their corresponding areas is a^2 : b^2 2.) the ratio of their volumes is a^3 : b^3

Converse of the Pythagorean Theorem

If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is ____.

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is OBTUSE.

If the square of the length of the longest side of a triangles is less than the sum of the squares of the lengths of the other two sides, the triangle is ____.

If the square of the length of the longest side of a triangles is less than the sum of the squares of the lengths of the other two sides, the triangle is ACUTE.

If three parallel lines intersect two transversals, then the segments intercepted on the transversals are ____.

If three parallel lines intersect two transversals, then the segments intercepted on the transversals are PROPORTIONAL.

Angle-Angle Similarity (AA~) Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Cavalieri's Principle

If two space figures have the same height and the same cross-sectional area at every level, then they have the same volume.

30-60-90 Triangle Theorem

In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is the square root of 3 times the length of the shorter leg.

45-45-90 Triangle Theorem

In a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is the square root of 2 times the length of a leg.

In a circle, a diameter that bisects a chord (that is not a diameter) is ____ to the chord.

In a circle, a diameter that bisects a chord (that is not a diameter) is PERPENDICULAR to the chord.

In a circle, a diameter that is perpendicular to a chord ____ the chord and its arcs.

In a circle, a diameter that is perpendicular to a chord BISECTS the chord and its arcs.

In a circle, the perpendicular bisector of a chord contains the ____ of the circle.

In a circle, the perpendicular bisector of a chord contains the CENTER of the circle.

Fundamental Theorem of Isometries

In a plane, one of two congruent figures can be mapped onto the other by a composition of at most three reflections.

Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Cosine (trigonometric ratio)

In right triangle ABC with acute angle <A cosine of <A = leg adjacent <A / hypotenuse

Sine (trigonometric ratio)

In right triangle ABC with acute angle <A sine of <A = leg opposite <A / hypotenuse

Tangent (trigonometric ratio)

In right triangle ABC with acute angle <A tangent of <A = leg opposite <A / leg adjacent <A

Using indirect measurement,

Indirect measurement is a way of measuring things that are difficult to measure directly.

Similar polygons

Polygons having corresponding angles congruent and corresponding sides proportional. *Denoted by ~*

Slant height

Regular pyramid: The length of an altitude of a lateral face. Right cone: The distance from the vertex to the edge of the base.

Given the height (altitude) and base of a pyramid or cone, how can you find the slant height?

Set up a right triangle with the height as one leg and the length of the base divided by two as the other. The hypotenuse is the slant height. Use Pythagorean Theorem to solve. h^2 + (b/2)^2 = slant height^2

Similar solids

Similar solids have the same shape and have all their corresponding dimensions proportional.

Trigonometric ratios with calculator

Sine: SIN = O/H, SIN^-1 to solve for angle measure Cosine: COS = A/H, COS^-1 to solve for angle measure Tangent: TAN = O/A, TAN^-1 to solve for angle measure *Calculator should be in degree mode*

Lateral area

Surface area minus base area(s) Prism or pyramid: The sum of the areas of the lateral faces Cylinder or cone: The area of the curved surface

The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are ____ to the original triangle and to each other.

The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are SIMILAR to the original triangle and to each other.

Angle of elevation

The angle formed by a horizontal line and the line of sight to an object above the horizontal line.

Angle of depression

The angle formed by a horizontal line and the line of sight to an object below the horizontal line.

Area of a parallelogram

The area of a parallelogram is the product of a base and the corresponding height. A = bh You can choose any side to be the base. An altitude is any segment perpendicular to the line containing the base drawn from the side opposite the base. The height is the length of an altitude.

Area of a regular polygon

The area of a regular polygon is half the product of the apothem and the perimeter. A = 1/2 aP

Area of a rhombus or kite

The area of a rhombus or a kite is half the product of the lengths of its diagonals. A = 1/2 d1 x d2

Area of a Sector of a Circle

The area of a sector of a circle is the product of the ratio (measure of the arc)/360 and the area of the circle.

Area of a trapezoid

The area of a trapezoid is half the product of the height and the sum of the bases. A = 1/2 h (b1+b2) The parallel sides are the bases. An altitude of a trapezoid is a perpendicular segment from one base to the line containing the other base. Its length is called the height of a trapezoid.

Area of a Triangle Given SAS

The area of a triangle is one half the product of the lengths of two sides and the sine of the included angle.

Composite space figure

The combination of two or more figures into one object.

Glide reflection

The composition of a translation followed by a reflection across a line parallel to the translation vector.

Apothem

The distance from the center to a side.

Radius of a regular polygon

The distance from the center to a vertex.

Preimage

The given figure a transformation is performed on.

Cross section

The intersection of a solid and a plane.

Arc Length

The length of an arc of a circle is the product of the ratio (measure of the arc)/360 and the circumference of the circle.

Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Arc Addition Postulate

The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Geometric mean

The number x such that a/x = x/b, where a, b, and x are positive numbers. x = square root of (ab)

Euler's Formula

The numbers of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula F+V = E+2.

The opposite angles of a quadrilateral inscribed in a circle are ____.

The opposite angles of a quadrilateral inscribed in a circle are SUPPLEMENTARY.

Segment of a circle

The part of a circle bounded by an arc and the segment joining its endpoints.

Cross-product property

The product of the extremes of a proportion is equal to the product of the means. Ex. If a/b = c/d, then ad = bc

Scale

The ratio of any length in a scale drawing to the corresponding actual length. *The lengths may be in different units*

Similarity ratio

The ratio of the lengths of corresponding sides of similar figures.

Sector of a circle

The region bounded by two radii and their intercepted arc.

Image

The resulting figure from a transformation.

Circle

The set of all points in a plane that are a given distance, the radius, from a given point, the center.

Sphere

The set of all points in space that are a given distance r, the radius, from a given point C, the center. A great circle is the intersection of a sphere with a plane containing the center of the sphere. The circumference of a sphere is the circumference of any great circle of the sphere.

Standard form of an equation of a circle

The standard form of an equation of a circle is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle.

Surface area

The sum of the lateral area and the areas of the bases.

The two segments tangent to a circle from a point outside the circle are ____.

The two segments tangent to a circle from a point outside the circle are CONGRUENT.

Glide reflectional symmetry

The type of symmetry for which there is a glide reflection that maps a figure onto itself.

Reflectional symmetry

The type of symmetry for which there is a reflection that maps a figure onto itself. The reflection line is the line of symmetry. Also called line symmetry

Rotational symmetry

The type of symmetry for which there is a rotation of 180 degrees or less that maps a figure onto itself.

Point symmetry

The type of symmetry for which there is a rotation of 180 degrees that maps a figure onto itself.

Translational symmetry

The type of symmetry for which there is a translation that maps a figure onto itself.

Isometry Classification Theorem

There are only four isometries 1.) Reflection 2.) Translation 3.) Rotation 4.) Glide reflection

Two inscribed angles that intercept the same arc are ____.

Two inscribed angles that intercept the same arc are CONGRUENT.

Within a circle or in congruent circles 1.) Chords equidistant from the center are ____. 2.) Congruent chords are ____ from the center.

Within a circle or in congruent circles 1.) Chords equidistant from the center are CONGRUENT. 2.) Congruent chords are EQUIDISTANT from the center.

Within a circle or in congruent circles 1.) Congruent central angles have ____ chords. 2.) Congruent chords have ____ arcs. 3.) Congruent arcs have ____ central angles.

Within a circle or in congruent circles 1.) Congruent central angles have CONGRUENT chords. 2.) Congruent chords have CONGRUENT arcs. 3.) Congruent arcs have CONGRUENT central angles.

Properties of proportions

a/b = c/d is equivalent to 1.) ad = bc 2.) b/a = d/c 3.) a/c = b/d 4.) (a+b)/b = (c+d)/d