Geometry IH Finals (Tucker)

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160 Terms

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What is the circumference of the circle?

The circumference C of a circle is C = πd or C = 2πr, where d is the diameter of the circle and r is the radius of the circle

<p>The circumference C of a circle is C = πd or C = 2πr, where d is the diameter of the circle and r is the radius of the circle</p>
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What is the arc length?

In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360 degrees

Arc length of ⌢AB / 2πr = m⌢AB/360

Arc length of ⌢AB = m⌢AB/360⋅2πr

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What is a radian?

For a circle with radius r, the measure of an angle in standard position whose terminal side intercepts an arc of length r is one radian

<p>For a circle with radius r, the measure of an angle in standard position whose terminal side intercepts an arc of length r is one radian</p>
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How do you convert degrees and radians?

Degrees to radians

Multiply degree measure by 2π radians/360 or πradians/180

Radians to degrees

Multiply radian measure by 360/2πradians, or 180/πradians

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What is the area of a circle?

The area of a circle is A = πr^2 where r is the radius of the circle

<p>The area of a circle is A = πr^2 where r is the radius of the circle</p>
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What is the area of a sector?

The ratio of the area of a sector of a circle to the area of the whole circle (πr^2) is equal to the ratio of the measure of the intercepted arc to 360

Area of sector APB /πr^2 = m⌢AB/360 or

Area of sector APB = m⌢AB/360⋅πr^2

<p>The ratio of the area of a sector of a circle to the area of the whole circle (πr^2) is equal to the ratio of the measure of the intercepted arc to 360</p><p>Area of sector APB /πr^2 = m⌢AB/360 or</p><p>Area of sector APB = m⌢AB/360⋅πr^2</p>
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What is the area of a rhombus or kite?

The area of a rhombus or kite is one-half the product of the diagonals d1 and d2

A = 1/2d1d2

<p>The area of a rhombus or kite is one-half the product of the diagonals d1 and d2</p><p>A = 1/2d1d2</p>
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What is the apothem of a regular polygon?

The distance from the center to any side of a regular polygon

<p>The distance from the center to any side of a regular polygon</p>
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What is the central angle of a regular polygon?

An angle formed by two radii drawn to consecutive vertices of a polygon

<p>An angle formed by two radii drawn to consecutive vertices of a polygon</p>
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What is the area of a regular polygon?

The area of a regular n-gon with side length s is one-half the product of the apothem a and the perimeter P

A = 1/2aP or A=1/2a⋅ns

<p>The area of a regular n-gon with side length s is one-half the product of the apothem a and the perimeter P</p><p>A = 1/2aP or A=1/2a⋅ns</p>
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What is a chord?

A segment whose endpoints are on a circle

<p>A segment whose endpoints are on a circle</p>
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What is a diameter?

A chord that contains the center of a circle

<p>A chord that contains the center of a circle</p>
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What is a secant?

A line that intersects a circle in two points

<p>A line that intersects a circle in two points</p>
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What is a tangent?

A line in the plane of a circle that intersects the circle in exactly one point, the point of tangency

<p>A line in the plane of a circle that intersects the circle in exactly one point, the point of tangency </p>
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What is the tangent line to circle theorem?

In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle

<p>In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle</p>
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What is the external tangent congruence theorem?

Tangent segments from a common external point are congruent

<p>Tangent segments from a common external point are congruent</p>
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What is a minor arc?

An arc with a measure less than 180 degrees

<p>An arc with a measure less than 180 degrees</p>
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What is a major arc?

An arc with a measure greater than 180 degrees

<p>An arc with a measure greater than 180 degrees</p>
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What is a semicircle?

An arc with endpoints that are endpoints of a diameter

<p>An arc with endpoints that are endpoints of a diameter</p>
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What is the measure of a minor arc?

The measure of its central angle

<p>The measure of its central angle</p>
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What is the measure of a major arc?

The difference of 360 degrees and the measure of the related minor arc

<p>The difference of 360 degrees and the measure of the related minor arc</p>
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What is the measure of a semicircle?

180 degrees

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What is the measure of a circle?

360 degrees

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What are adjacent arcs?

Arcs of a circle that have exactly one point in common

<p>Arcs of a circle that have exactly one point in common</p>
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What is the equidistant chords theorem?

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center

<p>In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center</p>
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What are congruent circles?

Circles that can be mapped onto each other by a rigid motion or a composition of rigid motions

<p>Circles that can be mapped onto each other by a rigid motion or a composition of rigid motions</p>
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What are congruent arcs?

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

<p>Arcs that have the same measure and are of the same circle or of congruent circles </p>
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What is the congruent circles theorem?

Two circles are congruent circles if and only if they have the same radius

<p>Two circles are congruent circles if and only if they have the same radius</p>
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What is the congruent central angles theorem?

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent

<p>In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent</p>
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What is the similar circles theorem?

All circles are similar

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What are similar arcs?

Arcs that have the same measure

<p>Arcs that have the same measure</p>
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What is the congruent corresponding chords theorem?

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent

<p>In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent</p>
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What is the perpendicular chord bisector theorem?

If a diameter of a circle is perpendicular to a chord then the diameter bisects the chord and its arc

<p>If a diameter of a circle is perpendicular to a chord then the diameter bisects the chord and its arc</p>
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What is the perpendicular chord bisector converse?

If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter

<p>If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter</p>
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What is the arc addition postulate?

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

<p>The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs</p>
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What is an inscribed angle?

An inscribed angle is an angle whose vertex lies on a circle and whose sides contain chords of the circle

<p>An inscribed angle is an angle whose vertex lies on a circle and whose sides contain chords of the circle</p>
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What is a intercepted arc?

An arc that lies between two lines, rays, or segments

<p>An arc that lies between two lines, rays, or segments</p>
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What does subtend mean?

If the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to subtend the angle

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What is the measure of an inscribed angle theorem?

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

<p>The measure of an inscribed angle is one-half the measure of its intercepted arc</p>
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What is an inscribed polygon?

A polygon is an inscribed polygon when all its vertices lie on a circle

<p>A polygon is an inscribed polygon when all its vertices lie on a circle</p>
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What is a circumscribed circle?

The circle that contains the vertices

<p>The circle that contains the vertices</p>
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What is the inscribed right triangle theorem?

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of a circle

Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle

<p>If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of a circle</p><p>Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle</p>
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What is the inscribed quadrilateral theorem?

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary

<p>A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary</p>
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What is the tangent and intersected chord theorem?

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is 1/2 the measure of its intercepted arc

<p>If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is 1/2 the measure of its intercepted arc</p>
45
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What are intersecting lines and circles?

If two nonparallel lines intersect a circle, there are three places where the lines can intersect

<p>If two nonparallel lines intersect a circle, there are three places where the lines can intersect</p>
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What is the angles inside the circle theorem?

If two chords intersect INSIDE a circle, then the measure of each angle is 1/2 the sum of the measures of the arcs intercepted by the angle and its vertical angle

<p>If two chords intersect INSIDE a circle, then the measure of each angle is 1/2 the sum of the measures of the arcs intercepted by the angle and its vertical angle</p>
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What is the angles outside the circle theorem?

If a tangent & a secant, two tangents, or two secants intersect OUTSIDE a circle, then the measure of the angle formed is 1/2 the difference of the measures of the intercepted arcs

<p>If a tangent &amp; a secant, two tangents, or two secants intersect OUTSIDE a circle, then the measure of the angle formed is 1/2 the difference of the measures of the intercepted arcs</p>
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What is the circumscribed angle theorem?

The measure of a circumscribed angle is equal to 180 degrees minus the measure of the central angle that intercepts the same arc

<p>The measure of a circumscribed angle is equal to 180 degrees minus the measure of the central angle that intercepts the same arc</p>
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What are segments of a chord?

The segments formed from two chords that intersect in the interior of a circle

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What is the segments of chords theorem?

If two chords intersect in the interior of a circle, then the products of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord

<p>If two chords intersect in the interior of a circle, then the products of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord</p>
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What is the tangent segment?

A tangent segment is a segment that is tangent to a circle at an endpoint

<p>A tangent segment is a segment that is tangent to a circle at an endpoint</p>
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What is a secant segment?

A segment that contains a chord of a circle and has exactly one endpoint outside the circle

<p>A segment that contains a chord of a circle and has exactly one endpoint outside the circle</p>
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What is a external segment?

The part of a secant segment that is outside the circle

<p>The part of a secant segment that is outside the circle</p>
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What is the segments of secants theorem?

If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment

<p>If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment</p>
55
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What is the segments of secants and tangents theorem?

If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment

<p>If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment</p>
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What is the standard equation of a circle?

(x-h)^2 + (y-k)^2 = r^2

Where r is the radius and (h,k) is the center

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What is the pythagorean theorem?

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

<p>In a right triangle, the square of the length of the hypotenuse is qual to the sum of the squares of the lengths of the legs</p>
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What is a pythagorean triple?

A set of three positive integers a, b, c that satisfy the equation - c^2 = a^2 + b^2

There can be no decimals or square roots

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What is the converse of the pythagorean theorem?

If the square of the length of the longest 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

<p>If the square of the length of the longest 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</p>
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What is the pythagorean inequalities theorem?

For any triangle ABC, where c is the length of the longest side the following statements are true

If c^2 < a^2 + b^2 then triangle ABC is acute

If c^2 > a^2 + b^2 then triangle ABC is obtuse

<p>For any triangle ABC, where c is the length of the longest side the following statements are true</p><p>If c^2 &lt; a^2 + b^2 then triangle ABC is acute</p><p>If c^2 &gt; a^2 + b^2 then triangle ABC is obtuse</p>
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What is the 45° - 45° - 90° triangle theorem?

In a 45° - 45° - 90° triangle, the hypotenuse is √2 times as long as each leg

Hypotenuse - leg x √2

<p>In a 45° - 45° - 90° triangle, the hypotenuse is √2 times as long as each leg</p><p>Hypotenuse - leg x √2</p>
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What is the 30° - 60° - 90° triangle theorem?

In a 30° - 60° - 90° triangle, the hypotenuse is twice as long as the shorter leg, and the long leg is √3 times as long as the shorter leg

Hypotenuse - shorter leg x 2

Longer leg = shorter leg x √3

<p>In a 30° - 60° - 90° triangle, the hypotenuse is twice as long as the shorter leg, and the long leg is √3 times as long as the shorter leg</p><p>Hypotenuse - shorter leg x 2</p><p>Longer leg = shorter leg x √3</p>
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What is the right triangle similarity theorem?

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other

<p>If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other</p>
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What is a geometric mean?

The geometric mean of two positive numbers a and b is the positive number x that satisfies a/x = x/b

So x^2 = ab and x = √ab

<p>The geometric mean of two positive numbers a and b is the positive number x that satisfies a/x = x/b</p><p>So x^2 = ab and x = √ab</p>
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What is the geometric mean (altitude) theorem?

In a right triangle, the altitude to the hypotenuse divides. the hypotenuse into two segments

The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse

<p>In a right triangle, the altitude to the hypotenuse divides. the hypotenuse into two segments</p><p>The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse</p>
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What is the geometric mean (leg) theorem?

In a right triangle, the altitude to the hypotenuse divides the hypotenuse into two segments

The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg

<p>In a right triangle, the altitude to the hypotenuse divides the hypotenuse into two segments</p><p>The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg</p>
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What is the trigonometric ratio?

A ratio of the lengths of two sides in a right triangle

Three most common are sine, cosine, and tangent

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What is tangent?

A trigonometric function for an acute angle of a right triangle - opposite over adjacent

<p>A trigonometric function for an acute angle of a right triangle - opposite over adjacent </p>
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What is the angle of elevation?

An angle formed by a horizontal line and a line of sight up to an object

<p>An angle formed by a horizontal line and a line of sight up to an object</p>
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What is sine?

A trigonometric function for an acute angle of a right triangle denoted by opposite over hypotenuse

<p>A trigonometric function for an acute angle of a right triangle denoted by opposite over hypotenuse</p>
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What is cosine?

A trigonometric function for an acute angle of a right triangle denoted by adjacent over hypotenuse

<p>A trigonometric function for an acute angle of a right triangle denoted by adjacent over hypotenuse </p>
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What is sine and cosine of complementary angles?

The sine of an acute angle is equal to the cosine of its complement

The cosine of an acute angle is equal to the sine of its complement

EX: sin A = cos(90-A) = cosB

<p>The sine of an acute angle is equal to the cosine of its complement</p><p>The cosine of an acute angle is equal to the sine of its complement</p><p>EX: sin A = cos(90-A) = cosB</p>
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What is the angle of depression?

An angle formed by a horizontal line and a line of sight down to an object

<p>An angle formed by a horizontal line and a line of sight down to an object</p>
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What is the inverse trigonometric ratios?

knowt flashcard image
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How do you solve a right triangle?

You must find all unknown side lengths and angle measures

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What is the area of a triangle?

The area of a triangle is given by one-half the product of the lengths of two sides times the sine of their included angle

<p>The area of a triangle is given by one-half the product of the lengths of two sides times the sine of their included angle</p>
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What is the Law of Sines?

It can be written in either of the following forms for triangle ABC with sides of length a, b, and c

sin A / a = sin B / b = sin C / c

or

a / sin A = b sin B = c / sin C

<p>It can be written in either of the following forms for triangle ABC with sides of length a, b, and c</p><p>sin A / a = sin B / b = sin C / c</p><p>or</p><p>a / sin A = b sin B = c / sin C</p>
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What is the Law if Cosines?

If triangle ABC has sides of length a, b, and c, as shown, then the following are true

a^2 - b^2 + c^2 -2bc cos A

b^2 = a^2 + c^2 - 2ac cos B

c^2 - a^2 + b^2 - 2ab cos C

<p>If triangle ABC has sides of length a, b, and c, as shown, then the following are true</p><p>a^2 - b^2 + c^2 -2bc cos A</p><p>b^2 = a^2 + c^2 - 2ac cos B</p><p>c^2 - a^2 + b^2 - 2ab cos C</p>
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What are corresponding lengths in similar polygons?

If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons

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What is the perimeters of similar polygons?

If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths

<p>If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths</p>
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What is the areas of similar polygons?

If two polygons are similar, then the ratio of their areas is equal to the squares of the ratios of their corresponding side lengths

<p>If two polygons are similar, then the ratio of their areas is equal to the squares of the ratios of their corresponding side lengths</p>
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What is the angle-angle (AA) similarity theorem?

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

<p>If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar</p>
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What is the side-side-side (SSS) similarity theorem?

If the corresponding side lengths of two triangles are proportional then the triangles are similar

<p>If the corresponding side lengths of two triangles are proportional then the triangles are similar</p>
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What is the side-angle-side (SAS) similarity theorem?

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

<p>If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar</p>
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What is the triangle proportionality theorem?

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

<p>If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally</p>
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What is the converse of triangle proportionality theorem?

If a line divides two sides of a triangle proportionality then it is parallel to the third side

<p>If a line divides two sides of a triangle proportionality then it is parallel to the third side</p>
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What is the three parallel lines theorem?

If three parallel lines intersect two transversals, then they divide the transversals proportionally

<p>If three parallel lines intersect two transversals, then they divide the transversals proportionally</p>
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What is the triangle angle bisector theorem?

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

<p>If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides</p>
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What is a similarity statement?

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What is a statement of proportionality?

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When looking for x in similar polygons, does it matter if x is the numerator or denominator?

No

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What is the def of a parallelogram?

If both pairs of opposite sides are parallel then the quadrilateral is a parallelogram

<p>If both pairs of opposite sides are parallel then the quadrilateral is a parallelogram</p>
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What is a diagonal?

A seg that joins two nonconsecutive vertices of a polygon

<p>A seg that joins two nonconsecutive vertices of a polygon</p>
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What is the polygon interior angles theorem?

The sum measures of the int. angles of a convex n-gon is (n-2)180

<p>The sum measures of the int. angles of a convex n-gon is (n-2)180</p>
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What is the corollary to the polygon int angles theorem?

The sum of measures of the int angles of a quadrilateral is 360 degrees

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What is an equilateral polygon?

A polygon with all sides congruent

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What is an equiangular polygon?

A polygon with all angles congruent

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What is a regular polygon?

A convex polygon that is both equilateral and equiangular

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What is the polygon exterior angles theorem?

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 degrees

<p>The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 degrees</p>
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What is the parallelogram opposite sides theorem?

If a quadrilateral is a parallelogram, then its opposite sides are congruent

<p>If a quadrilateral is a parallelogram, then its opposite sides are congruent</p>