Home
Bookshelves
Pre-Algebra
Prealgebra 1e (OpenStax)
9: Math Models and Geometry
9.5: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 2)

9.5: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 2)

Last updated
Save as PDF

Page ID: 7024

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Use the Pythagorean Theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around 500 BCE.

Remember that a right triangle has a 90° angle, which we usually mark with a small square in the corner. The side of the triangle opposite the 90° angle is called the hypotenuse, and the other two sides are called the legs. See Figure $\PageIndex{8}$.

$Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.”$

Figure $\PageIndex{8}$ - In a right triangle, the side opposite the 90° angle is called the hypotenuse and each of the other sides is called a leg.

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.

Definition: The Pythagorean Theorem

In any right triangle ΔABC,

\[a^{2} + b^{2} = c^{2}\]

where c is the length of the hypotenuse a and b are the lengths of the legs.

$A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.$

To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation m and defined it in this way:

\[If\; m = n^{2}, then\; \sqrt{m} = n\; for\; n \geq 0\]

For example, we found that $\sqrt{25}$ is 5 because 5² = 25.

We will use this definition of square roots to solve for the length of a side in a right triangle.

Example $\PageIndex{7}$:

Use the Pythagorean Theorem to find the length of the hypotenuse.

$Right triangle with legs labeled as 3 and 4.$

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length of the hypotenuse of the triangle
Step 3. Name. Choose a variable to represent it.	Let c = the length of the hypotenuse $CNX_BMath_Figure_09_03_025.jpg$
Step 4. Translate. Write the appropriate formula. Substitute.	$$\begin{split} a^{2} + b^{2} &= c^{2} \\ 3^{2} + 4^{2} &= c^{2} \end{split}$$
Step 5. Solve the equation.	$$\begin{split} 9 + 16 &= c^{2} \\ 25 &= c^{2} \\ \sqrt{25} &= c^{2} \\ 5 &= c \end{split}$$
Step 6. Check.	$$\begin{split} 3^{2} + 4^{2} &= \textcolor{red}{5}^{2} \\ 9 + 16 &\stackrel{?}{=} 25 \\ 25 &= 25\; \checkmark \end{split}$$
Step 7. Answer the question.	The length of the hypotenuse is 5.

Exercise $\PageIndex{13}$:

Use the Pythagorean Theorem to find the length of the hypotenuse.

$A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.$

Answer: c = 10

Exercise $\PageIndex{14}$:

Use the Pythagorean Theorem to find the length of the hypotenuse.

$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled “8”.$

Answer: c = 17

Example $\PageIndex{8}$:

Use the Pythagorean Theorem to find the length of the longer leg.

$Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.$

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	The length of the leg of the triangle
Step 3. Name. Choose a variable to represent it.	Let b = the leg of the triangle Label side b $CNX_BMath_Figure_09_03_054_img-01.png$
Step 4. Translate. Write the appropriate formula. Substitute.	$$\begin{split} a^{2} + b^{2} &= c^{2} \\ 5^{2} + b^{2} &= 13^{2} \end{split}$$
Step 5. Solve the equation.	$$\begin{split} 25 + b^{2} &= 169 \\ b^{2} &= 144 \\ b^{2} &= \sqrt{144} \\ b &= 12 \end{split}$$
Step 6. Check.	$$\begin{split} 5^{2} + \textcolor{red}{12}^{2} &\stackrel{?}{=} 13^{2} \\ 25 + 144 &\stackrel{?}{=} 169 \\ 169 &= 169\; \checkmark \end{split}$$
Step 7. Answer the question.	The length of the leg is 12.

Exercise $\PageIndex{15}$:

Use the Pythagorean Theorem to find the length of the leg.

$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled “b”.$

Answer: b = 8

Exercise $\PageIndex{16}$:

Use the Pythagorean Theorem to find the length of the leg.

$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled “b”.$

Answer: b = 12

Example $\PageIndex{9}$:

Kelvin is building a gazebo and wants to brace each corner by placing a 10-inch wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

$A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.$

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let x = the distance from the corner $CNX_BMath_Figure_09_03_055_img-01.png$
Step 4. Translate. Write the appropriate formula. Substitute.	$$\begin{split} a^{2} + b^{2} &= c^{2} \\ x^{2} + x^{2} &= 10^{2} \end{split}$$
Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth.	$$\begin{split} 2x^{2} &= 100 \\ x^{2} &= 50 \\ x &= \sqrt{50} \\ b &\approx 7.1 \end{split}$$
Step 6. Check.	$$\begin{split} a^{2} + b^{2} &= c^{2} \\ (\textcolor{red}{7.1})^{2} + (\textcolor{red}{7.1})^{2} &\stackrel{?}{\approx} 10^{2} \\ &Yes. \end{split}$$
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

Exercise $\PageIndex{17}$:

John puts the base of a 13-ft ladder 5 feet from the wall of his house. How far up the wall does the ladder reach?

$A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.$

Answer: 12 feet

Exercise $\PageIndex{18}$:

Randy wants to attach a 17-ft string of lights to the top of the 15-ft mast of his sailboat. How far from the base of the mast should he attach the end of the light string?

$A picture of a boat is shown. The height of the center pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.$

Answer: 8 feet

ACCESS ADDITIONAL ONLINE RESOURCES

Animation: The Sum of the Interior Angles of a Triangle

Similar Polygons

Example: Determine the Length of the Hypotenuse of a Right Triangle

Practice Makes Perfect

Use the Properties of Angles

In the following exercises, find (a) the supplement and (b) the complement of the given angle.

53°
16°
29°
72°

In the following exercises, use the properties of angles to solve.

Find the supplement of a 135° angle.
Find the complement of a 38° angle.
Find the complement of a 27.5° angle.
Find the supplement of a 109.5° angle.
Two angles are supplementary. The larger angle is 56° more than the smaller angle. Find the measures of both angles.
Two angles are supplementary. The smaller angle is 36° less than the larger angle. Find the measures of both angles.
Two angles are complementary. The smaller angle is 34° less than the larger angle. Find the measures of both angles.
Two angles are complementary. The larger angle is 52° more than the smaller angle. Find the measures of both angles.

Use the Properties of Triangles

In the following exercises, solve using properties of triangles.

The measures of two angles of a triangle are 26° and 98°. Find the measure of the third angle.
The measures of two angles of a triangle are 61° and 84°. Find the measure of the third angle.
The measures of two angles of a triangle are 105° and 31°. Find the measure of the third angle.
The measures of two angles of a triangle are 47° and 72°. Find the measure of the third angle.
One angle of a right triangle measures 33°. What is the measure of the other angle?
One angle of a right triangle measures 51°. What is the measure of the other angle?
One angle of a right triangle measures 22.5°. What is the measure of the other angle?
One angle of a right triangle measures 36.5°. What is the measure of the other angle?
The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.
The measure of the smallest angle of a right triangle is 20° less than the measure of the other small angle. Find the measures of all three angles.
The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.
The angles in a triangle are such that the measure of one angle is 20° more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

Find the Length of the Missing Side

In the following exercises, Δ ABC is similar to Δ XYZ. Find the length of the indicated side.

$Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 9, the side across from B is labeled b, and the side across from C is labeled 15. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled 8, and the side across from Z is labeled 10.$

side b
side x

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is 270 miles.

$A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.$

Find the distance from Los Angeles to San Francisco.
Find the distance from San Francisco to Las Vegas.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

$A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.$
$A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.$
$A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.$
$A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.$

Find the Length of the Missing Side

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.$
$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.$
$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.$
$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.$
$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.$
$A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.$
$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.$
$A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.$

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

A 13-foot string of lights will be attached to the top of a 12-foot pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?

$A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.$

Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is 12 feet high and 16 feet wide. How long should the banner be to fit the garage door?

$A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.$

Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of 10 feet. What will the length of the path be?

$A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.$

Brian borrowed a 20-foot extension ladder to paint his house. If he sets the base of the ladder 6 feet from the house, how far up will the top of the ladder reach?

$A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.$

Everyday Math

Building a scale model Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is 30 feet wide and 35 feet tall at the highest point of the roof. If the dollhouse will be 2.5 feet wide, how tall will its highest point be?
Measurement A city engineer plans to build a footbridge across a lake from point X to point Y, as shown in the picture below. To find the length of the footbridge, she draws a right triangle XYZ, with right angle at X. She measures the distance from X to Z, 800 feet, and from Y to Z, 1,000 feet. How long will the bridge be?

$A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.$

Writing Exercises

Write three of the properties of triangles from this section and then explain each in your own words.
Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length 3 and 4.

$Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.$

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

$CNX_BMath_Figure_AppB_053.jpg$

(b) What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."

Search

Text Color

Text Size

Margin Size

Font Type

Definition: The Pythagorean Theorem

Example \(\PageIndex{7}\):

Exercise \(\PageIndex{13}\):

Exercise \(\PageIndex{14}\):

Example \(\PageIndex{8}\):

Exercise \(\PageIndex{15}\):

Exercise \(\PageIndex{16}\):

Example \(\PageIndex{9}\):

Exercise \(\PageIndex{17}\):

Exercise \(\PageIndex{18}\):

ACCESS ADDITIONAL ONLINE RESOURCES