Geometry Honors Second Semester Final Exam Study Guide

Area of Sector

(m"arc"/360)×πr²

*Chapter 11*

*Area*

*Chapter 10*

*Circles*

*Chapter 8*

*Similar Polygons*

*Chapter 12*

*Surface Area and Volume*

*Chapter 9*

*The Pythagorean Theorem*

Arc Length

-A fraction of the circle's circumference. -((mAB/360⁰)×360⁰)÷2πr

Radius Chord Relationships (Sect. 10.1)

-If a radius is ⊥ to a chord ,then it bisects the chord. -If a radius of a circle bisects a chord that is not a diameter, then it is ⊥ to that chord. -The ⊥ bisector of a chord passes through the center of the circle.

Altitude on Hypotenuse Theorems

-If an altitude is drawn to the hypotenuse of a right ∆, the 2 ∆s formed are similar to the given right ∆ and to each other. -If an altitude is drawn to the hypotenuse of a right ∆, the altitude to the hypotenuse is the geometric mean between the segments of the hypotenuse (altitude/2nd segment of hypotenuse=1st segment of hypotenuse/altitude). -If an altitude is drawn to the hypotenuse of a right ∆, either leg of a given right ∆ is the geometric mean between the hypotenuse of the given right ∆ and the segment of the hypotenuse adjacent to that leg (hypotenuse/leg=leg/segment of hypotenuse adjacent to leg).

Walk around problems

-THE HUNNY THEOREM (sum of two opposite sides of a quadrilateral=sum of the other two opposite sides of the quadrilateral.....only works for quadrilaterals, not ∆s). -set one part of a segment as x and "walk around" the figure, using the given information and the Two-Tangent Theorem (If two tangent segments are drawn to a circle from an external point, then those segments are congruent.).

Triangle Similarity (Theorems and Postulates)

-The ratio of the perimeters of two similar polygons equals the ration of any pair of corresponding sides. -If there exists a correspondence between the vertices of two ∆s such that the three ∠s of 1 ∆ are ≅ to the corresponding ∠s of the other triangle, then the ∆s are similar. (AAA) -If there exists a correspondence between the vertices of 2 ∆s such that 2 ∠s of 1 ∆ are congruent to the corresponding ∠s of the other, then the ∆s are similar. (AA) -If there exists a correspondence between the vertices of 2 ∆s such that the ratios of the measures of the corresponding sides are equal, then the ∆s are similar. (SSS∼) -If there exists a correspondence between the vertices of 2 ∆s such that the ratios of the measures of 2 pairs of corresponding sides are equal and the included ∠s are congruent, then the ∆s are similar. (SAS∼)

Ratio of perimeters and ratio of areas of similar polygons

-The ratio of the perimeters of two similar polygons equals the ration of any pair of corresponding sides. -If two figures are similar, then the ratio of their areas equals the square of the ratio of corresponding segments. (Similar-Figures Theorem) A₁/A₂=(s₁/s₂)².

Measure of Central Angles

-an ∠ whose vertex is at the center of the circle. -equals the measure of the arc that it intercepts.

Angle of elevation/depression

-angle of elevation=the ∠between an upward line of sight and the horizontal. -angle of depression= the ∠ between a downward line of sight and the horizontal.

Similarity ratios of similar polygons

-figures that have the same shape but not necessarily the same size are called similar figures -the ratios of the measures of corresponding sides of similar figures are equal -corresponding ∠s of similar figures are ≅ -∆ABC∼∆DEF ("∼" means "is similar to")

Solving missing sides and angles using trig functions

-sin(θ)=opposite/hypotenuse -cos(θ)=adjacent/hypotenuse -tan(θ)=opposite/adjacent

Proportions

-ways to write a ratio= a/b, a:b, a to b, a ÷ b -in a/b=c/d, a is the 1st term, b is the 2nd, c is the 3rd, and d is the fourth term (1st and 4th terms are called the extremes, 2nd and 3rd terms are called the means) -in a mean proportion, the means are the same (1/3=3/9), and the product of the means equals the products of the extremes

Area of Trapezoid

-½h(b₁+b₂) -Mh

Common Tangent Procedure

1. Draw the segment joining the centers. 2. Draw the radii to the points of contact. 3. Through the center of the smaller circle, draw a line parallel to the common tangent. 4. Observe that this line will intersect the radius of the larger circle (extended if necessary) to form a rectangle and a right ∆. 5. Use the Pythagorean Theorem and properties of a rectangle.

SA of Sphere

4πr² (4 × π × radius²)

Area of Parallelogram

A=bh

3 parallel lines cut by a transversal

If 3 or more parallel lines are intersected by 2 transversals, the parallel lines divide the transversal proportionally.

Side-Splitter Theorem

If a line is parallel to one side of a ∆ and intersects the other 2 sides, then it divides those 2 sides proportionally.

Angle Bisector Theorem

If a ray bisects an ∠ of a ∆, it divides the opposite side into segments that are proportional to the adjacent sides.

Tangent-Secant Power Theorem

If a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the measure of the tangent segment is equal to the product of the measures of the entire secant segment and its external part.

Classifying triangles given their side lengths

If c is the length of the longest side of a ∆ and: -c²<a²+b², then the ∆ is acute. -c²=a²+b², then the ∆ is right. -c²>a²+b², then the ∆ is obtuse.

Chord-Chord Power Theorem

If two chords of a circle intersect inside the circle, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord.

Secant-Secant Power Theorem

If two secant segments are drawn from an external point to a circle, then the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part.

30⁰-60⁰-90⁰ Triangles

In a ∆ whose ∠s have the measures 30⁰, 60⁰, and 90⁰, the lengths of the sides opposite these angles can be represented by x, x√3, and 2x, respectively (30⁰-60⁰-90⁰ Theorem).

45⁰-45⁰-90⁰ Triangles

In a ∆ whose ∠s have the measures 45⁰-45⁰-90⁰, the lengths of the sides opposite these ∠s can be represented by x, x, and x√2, respectively (45⁰-45⁰-90⁰ Theorem).

Volume of Prism, Cylinder, Pyramid, Cone, Sphere, and Composite Figure

Prism -Bh (area of base × height). Cylinder -πr²h (π × radius² × height). Pyramid -1/3Bh (1/3 × area of base × height). Cone -1/3πr²h (1/3 × π × radius² × height). Sphere -4/3πr³ (4/3 × π × radius³).

LA and SA of Prism, Cylinder, Pyramid, and Cone

Prism -The lateral surface area of a prism is the sum of the areas of the lateral faces. -The total surface area of a prism is the sum of the prism's lateral area and the areas of the two bases. -Area of any prism= 2B + Ph (2 × area of base + perimeter of the base × height of the prism). -Perimeter of any prism= Ph (perimeter of base × height of the prism). Cylinder -LA=2πrh -TA=LA+2A (2 × area of base). Pyramid -LA= ½Pl (½ perimeter of base × slant height). -TA= LA + B (area of base). Cone -LA= πrl (π × radius × slant height). -TA= LA + πr² (π × radius²).

Measure of Chord-Chord Angles

The measure of a chord-chord ∠ is one-half the sum of the measures of the arcs intercepted by the chord-chord ∠ and its vertical ∠.

Measure of Secant-Secant Angles

The measure of a secant-secant ∠, a secant-tangent ∠, or a tangent-tangent ∠ (vertex outside a circle) is one-half the difference of the measures of the intercepted arcs.

Measure of Tangent-Tangent Angles

The measure of a secant-secant ∠, a secant-tangent ∠, or a tangent-tangent ∠ (vertex outside a circle) is one-half the difference of the measures of the intercepted arcs.

Measure of Inscribed Angles

The measure of an inscribed ∠ or a tangent-chord ∠ (vertex on a circle) is one-half the measure of its intercepted arc.

Pythagorean Theorem

The square of the measure of the hypotenuse of a right ∆ is equal to the sum of the squares of the measures of the legs (a²+b²=c²).

Area of Segment

area of sector−area of ∆ inside sector

Distance Formula

d=√(x₁−x₂)²+(y₁−y₂)²

Area of Polygon

½ap

Area of Triangle

½bh

Area of Rhombus

½d₁d₂