# 3.4 Solve geometry applications: triangles, rectangles, and the  (Page 4/8)

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Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

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

## 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. Label side c on the figure. Let c = the length of the hypotenuse. Step 4. Translate. Write the appropriate formula. ${a}^{2}+{b}^{2}={c}^{2}$ Substitute. ${3}^{2}+{4}^{2}={c}^{2}$ Step 5. Solve the equation. $\phantom{\rule{0.4em}{0ex}}9+16={c}^{2}$ Simplify. $\phantom{\rule{2em}{0ex}}25={c}^{2}$ Use the definition of square root. $\phantom{\rule{1.5em}{0ex}}\sqrt{25}=c$ Simplify. $\phantom{\rule{2.5em}{0ex}}5=c$ Step 6. Check. Step 7. Answer the question. The length of the hypotenuse is 5.

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

$c=10$

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

$c=13$

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

## 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. Lable side b . Step 4. Translate Write the appropriate formula. ${a}^{2}+{b}^{2}={c}^{2}$ Substitute. ${5}^{2}+{b}^{2}={13}^{2}$ Step 5. Solve the equation. $25+{b}^{2}=169$ Isolate the variable term. $\phantom{\rule{2.1em}{0ex}}{b}^{2}=144$ Use the definition of square root. $\phantom{\rule{2.1em}{0ex}}{b}^{2}=\sqrt{144}$ Simplify. $\phantom{\rule{2.6em}{0ex}}b=12$ Step 6. Check. Step 7. Answer the question. The length of the leg is 12.

Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

8

Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

12

Kelvin is building a gazebo and wants to brace each corner by placing a $10\text{″}$ piece of wood diagonally as shown above.

If he fastens the wood so that the ends of the brace are the same distance from the corner, what is the length of the legs of the right triangle formed? Approximate to the nearest tenth of an inch.

## Solution

$\begin{array}{cccc}\mathbf{\text{Step 1.}}\phantom{\rule{0.2em}{0ex}}\text{Read the problem.}\hfill & & & \\ \mathbf{\text{Step 2.}}\phantom{\rule{0.2em}{0ex}}\text{Identify what we are looking for.}\hfill & & & \begin{array}{c}\text{the distance from the corner that the}\hfill \\ \text{bracket should be attached}\hfill \end{array}\hfill \\ \\ \\ \mathbf{\text{Step 3.}}\phantom{\rule{0.2em}{0ex}}\text{Name. Choose a variable to represent it.}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}x=\text{the distance from the corner.}\hfill \\ \\ \\ \begin{array}{c}\mathbf{\text{Step 4.}}\phantom{\rule{0.2em}{0ex}}\text{Translate.}\hfill \\ \text{Write the appropriate formula and substitute.}\hfill \\ \\ \\ \mathbf{\text{Step 5.}}\phantom{\rule{0.2em}{0ex}}\text{Solve the equation.}\hfill \\ \\ \phantom{\rule{2.5em}{0ex}}\text{Isolate the variable.}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{Use the definition of square root.}\hfill \\ \phantom{\rule{2.5em}{0ex}}\text{Simplify. Approximate to the nearest tenth.}\hfill \end{array}\hfill & & & \hfill \begin{array}{}\\ \hfill {a}^{2}+{b}^{2}& =\hfill & {c}^{2}\hfill \\ \hfill {x}^{2}+{x}^{2}& =\hfill & {10}^{2}\hfill \\ \\ \\ \hfill 2{x}^{2}& =\hfill & 100\hfill \\ \hfill {x}^{2}& =\hfill & 50\hfill \\ \hfill x& =\hfill & \sqrt{50}\hfill \\ \hfill x& \approx \hfill & 7.1\hfill \end{array}\hfill \\ \mathbf{\text{Step 6.}}\phantom{\rule{0.2em}{0ex}}\text{Check.}\hfill & & & \\ \begin{array}{ccc}\hfill \phantom{\rule{2.5em}{0ex}}{a}^{2}+{b}^{2}& =\hfill & {c}^{2}\hfill \\ \hfill \phantom{\rule{2.5em}{0ex}}{\left(7.1\right)}^{2}+{\left(7.1\right)}^{2}& \approx \hfill & {10}^{2}\phantom{\rule{0.2em}{0ex}}\text{Yes.}\hfill \end{array}\hfill & & & \\ \mathbf{\text{Step 7.}}\phantom{\rule{0.2em}{0ex}}\text{Answer the question.}\hfill & & & \begin{array}{c}\text{Kelvin should fasten each piece of}\hfill \\ \text{wood approximately}\phantom{\rule{0.2em}{0ex}}7.1\text{″}\phantom{\rule{0.2em}{0ex}}\text{from the corner.}\hfill \end{array}\hfill \end{array}$

John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. How far up the wall does the ladder reach?

12 feet

Randy wants to attach a 17 foot string of lights to the top of the 15 foot mast of his sailboat, as shown below. How far from the base of the mast should he attach the end of the light string?

8 feet

## Solve applications using rectangle properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right $\left(90\text{°}\right)$ angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L , and its adjacent side as the width, W .

#### Questions & Answers

A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
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Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot? Bridget Reply The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water. Bridget help me understand graphs Marlene Reply what kind of graphs? bruce function f(x) to find each value Marlene I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much. Marlene Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6 Melissa if they say find f(5) then the answer would be f(5)=12 Melissa I need you to help me Melissa. Wish I can show you my homework Marlene How is f(1) =0 I am really confused Marlene what's the formula given? f(x)=? Melissa It shows a graph that I wish I could send photo of to you on here Marlene Which problem specifically? Melissa which problem? Melissa I don't know any to be honest. But whatever you can help me with for I can practice will help Marlene I got it. sorry, was out and about. I'll look at it now. Melissa Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things. Marlene I just responded. Melissa Thank you Marlene -65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r WENDY Reply State the question clearly please Rich write in this form a/b answer should be in the simplest form 5% August Reply convert to decimal 9/11 August 0.81818 Rich 5/100 = .05 but Rich is right that 9/11 = .81818 Melissa Equation in the form of a pending point y+2=1/6(×-4) Jose Reply write in simplest form 3 4/2 August definition of quadratic formula Ahmed Reply From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots. Melissa what is the answer of w-2.6=7.55 What Reply 10.15 Michael w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6) Korin Nataly is considering two job offers. The first job would pay her$83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first? Mckenzie Reply x >$110,000
bruce
greater than \$110,000
Michael