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

 Page 4 / 8

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

Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply will every polynomial have finite number of multiples? cricket Reply a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check.$740+$170=$910.
David Reply
. A cashier has 54 bills, all of which are $10 or$20 bills. The total value of the money is $910. How many of each type of bill does the cashier have? jojo Reply whats the coefficient of 17x Dwayne Reply the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong Dwayne 17 Melissa wow the exercise told me 17x solution is 14x lmao Dwayne thank you Dwayne 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 Mikaela Reply Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself? Sam Reply Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers. Mckenzie Reply 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? Reiley Reply 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? hamzzi Reply 90 minutes muhammad 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 Nakiya Reply 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

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