Geometry Chapter 9 Study Guide

Theorem 68 Part C

If an altitude is drawn to the hypotenuse of a right triangle, then either leg of the given right triangle is the mean proportional between the hypotenuse adjacent to that leg.

Theorem 68 Part B

If an altitude is drawn to the hypotenuse of a right triangle, then the altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse.

Theorem 68 Part A

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

Converse of Pythagorean Theorem

If the square of the measure of one side of a triangle equals the sum of the squares of the measures of the other two sides, then the angle opposite the longest side is a right angle.

Mid Point Theorem

Square root of ((X2 - X1) squared + (Y2 - Y1)) squared

Simplified Radical Form

The most simple form of a radical.

Measure

The number of degrees it occupies comparatively to a circle. (360 degrees in a circle)

Cos of an Right Triangle

The ratio of the adjacent side and given angle divided by the hypotenuse

Tan of an Right Triangle

The ratio of the opposite side and given angle divided by the adjacent side

Sin of Right Triangle

The ratio of the opposite side and given angle divided by the hypotenuse

Circle

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

Circumference Formula

pi • diameter

Formula For The Area of a Circle

pi • radius squared

Quadratic Formula

(-b + or - radical (b squared - 4ac)) / 2

Length Formula

(measure of arc / 360) • circumference

Diameter

A cord that passes through the center of a circle. (longest cord)

Length

A fraction or portion of a circles circumference.

Radius

A segment with endpoints on the circle and the center point.

How to Find Obtuse Triangle (Pythagorean Theorem)

A squared + B squared < C squared

How to Find Right Triangle (Pythagorean Theorem)

A squared + B squared = C squared

How to Find Acute Triangle (Pythagorean Theorem)

A squared + B squared > C squared

Vocabulary for Regular Square Pyramids: Base + # Vertex + # Altitude + # Lateral Edge + # Slant Height

Base: Bottom square plane (1) Vertex: The point at the top (1) Altitude: Segment perpendicular to the base at its center (1) Lateral Edge: The edge of the pyramid formed by the faces of the pyramid (4) Slant Height: Segmant perpendicular to a side of the base

Pythagorean Theorem

If A squared + B squared = C squared than the triangle is right

Circumference

The length around the entire circle.

Using Families of Triangles to Shorten Right Triangle Computation

1. Reduce the difficulty of the problem by multiplying or dividing the three lengths by the same number to obtain a similar, but simpler triangle 2. Solve for the missing side of the easier triangle 3. Convert back to the origanal problem

How to Simplify rads

1. Turn the rad into multiples of itself, both being rads with one being a perfect square 2. solve the smaller rad that is a perfect square 3. put the solved rad in front of the unsolved rad 4. repeat till the rad is not able to be simplified (each time you get a solved rad multiply the coefficients)

Area

A quantity that expresses the extent of a two-dimensional surface or shape in the plane.

Inscribed angle

An angle whose vertex is on a circle and whose sides are determined by two chords.

Intercepted Arc

An arc whose endpoints are on the sides of an angle whose other points all line within the angle

Pythagorean Triple

Any three whole numbers that satisfy Pythagorean theorem: 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17

Sector

Area of the circle that is bounded by two radii and an arc of the circle.

Vocabulary for Rectangular Solid: Faces + # Edges + # Diagonals + #

Faces: Surface planes of the solid (6) Edges: Intersection of the faces (12) Diagonals: Segments connecting the opposite ends of the solid, through the solid (4)

30 - 60 - 90 Triangle

In a triangle whose angles have the measures 30, 60 and 90, the length of the sides opposite these angles can be represented by X, X rad3, and 2X respectively Hypotenuse = 2X 90 and 30 = X rad3 60 and 90 = X

45 - 45 - 90 Triangle

In a triangle whose angles have the measures 45, 45 and 90, the length of the sides opposite these angles can be represented by X, X rad2, and X respectively Hypotenuse = X rad2 Both 45 and 90 = X

Chord

Line segment joining two points on a circle.

Arcs

Made up of two points on a circle and all the points of the circle needed to connect those two points by a single path. Represented with an arc above the endpoints.

Angle of Depression

The angle that shows depression (going down) between the line drawn with two given points

Angle of Elevation

The angle that shows elevation (going up) between the line drawn with two given points

Does the altitude of an equilateral triangle bisects the opposite side?

Yes, proved with HL

Applying Right Triangles to Diagonals in a Rectangular Prism

You can use right triangles to find diagonals of rectangular prisms