# 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 .

Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
26 + 37 = 63 + 8 = 71
gayla
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned$1600 more than Jim earned. How much did they earned
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike
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?
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
how to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0