8.8: Chapter 8 Review
Keywords & Theorems
- Pythagorean Theorem
- Given a right triangle with legs of lengths \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and a hypotenuse of length \begin{align*}c\end{align*}, then \begin{align*}a^2 + b^2 = c^2\end{align*}.
- Pythagorean Triple
- A set of three whole numbers that makes the Pythagorean Theorem true.
- Distance Formula
- \begin{align*}d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\end{align*}.
- Pythagorean Theorem Converse
- If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
- Theorem 8-3
- If the sum of the squares of the two shorter sides in a right triangle is greater than the square of the longest side, then the triangle is acute.
- Theorem 8-4
- If the sum of the squares of the two shorter sides in a right triangle is less than the square of the longest side, then the triangle is obtuse.
- Theorem 8-5
- If an altitude is drawn from the right angle of any right triangle, then the two triangles formed are similar to the original triangle and all three triangles are similar to each other.
- Geometric Mean
- The geometric mean is a different sort of average, which takes the \begin{align*}n^{th}\end{align*} root of the product of \begin{align*}n\end{align*} numbers.
- Theorem 8-6
- In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of these two segments.
- Theorem 8-7
- In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments
- 45-45-90 Corollary
- If a triangle is an isosceles right triangle, then its sides are in the extended ratio \begin{align*}x : x : x \sqrt{2}\end{align*}.
- 30-60-90 Corollary
- If a triangle is a 30-60-90 triangle, then its sides are in the extended ratio \begin{align*}x : x \sqrt{3} : 2x\end{align*}.
- Trigonometry
- The study of the relationships between the sides and angles of right triangles.
- Adjacent (Leg)
- A side adjacent to an angle is the leg of the triangle that helps form the angle.
- Opposite (Leg)
- A side opposite to an angle is the leg of the triangle that does not help form the angle.
- Sine Ratio
- For an acute angle \begin{align*}x\end{align*} in a right triangle, the \begin{align*}\sin x\end{align*} is equal to the ratio of the side opposite the angle over the hypotenuse of the triangle.
- Cosine Ratio
- For an acute angle \begin{align*}x\end{align*} in a right triangle, the \begin{align*}\cos x\end{align*} is equal to the ratio of the side adjacent to the angle over the hypotenuse of the triangle.
- Tangent Ratio
- For an acute angle \begin{align*}x\end{align*}, in a right triangle, the \begin{align*}\tan x\end{align*} is equal to the ratio of the side opposite to the angle over the side adjacent to \begin{align*}x\end{align*}.
- Angle of Depression
- The angle measured from the horizon or horizontal line, down.
- Angle of Elevation
- The angle measure from the horizon or horizontal line, up.
- Inverse Tangent
- Inverse tangent is also called arctangent and is labeled \begin{align*}\tan^{-1}\end{align*} or arctan. The “-1” indicates inverse.
- Inverse Sine
- Inverse sine is also called arcsine and is labeled \begin{align*}\sin^{-1}\end{align*} or arcsin.
- Inverse Cosine
- Inverse cosine is also called arccosine and is labeled \begin{align*}\cos^{-1}\end{align*} or arccos.
- Law of Sines
- If \begin{align*}\triangle ABC\end{align*} has sides of length, \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*}, then \begin{align*}\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\end{align*}.
- Law of Cosines
- If \begin{align*}\triangle ABC\end{align*} has sides of length \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*}, then \begin{align*}a^2 = b^2 + c^2 - 2bc \cos A\end{align*}
\begin{align*}b^2 & = a^2 + c^2 - 2ac \ \cos B\\ c^2 & = a^2 + b^2 - 2ab \ \cos C\end{align*}
Review Questions
Solve the following right triangles using the Pythagorean Theorem, the trigonometric ratios, and the inverse trigonometric ratios. When possible, simplify the radical. If not, round all decimal answers to the nearest tenth.
Determine if the following lengths make an acute, right, or obtuse triangle. If they make a right triangle, determine if the lengths are a Pythagorean triple.
- 11, 12, 13
- 16, 30, 34
- 20, 25, 42
- \begin{align*}10 \sqrt{6}, 30, 10 \sqrt{15}\end{align*}
- 22, 25, 31
- 47, 27, 35
Find the value of \begin{align*}x\end{align*}.
- The angle of elevation from the base of a mountain to its peak is \begin{align*}76^\circ\end{align*}. If its height is 2500 feet, what is the distance a person would climb to reach the top? Round your answer to the nearest tenth.
- Taylor is taking an aerial tour of San Francisco in a helicopter. He spots AT&T Park (baseball stadium) at a horizontal distance of 850 feet and down (vertical) 475 feet. What is the angle of depression from the helicopter to the park? Round your answer to the nearest tenth.
Use the Law of Sines and Cosines to solve the following triangles. Round your answers to the nearest tenth.
Texas Instruments Resources
In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9693.