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Use the Pythagorean Theorem

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

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

In the following exercises, solve using the Pythagorean Theorem. 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, as shown below. How far from the base of the pole should the end of the string of lights be anchored?

A right triangle with one leg marked 12 and hypotenuse marked 13.

5 feet

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Pam wants to put a banner across her garage door, as shown below, to congratulate her son for 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 house is shown with a banner over the garage door. The garage door is marked 16 ft wide and 12 ft high.
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Chi is planning to put a path of paving stones through her flower garden, as shown below. The flower garden is a square with side 10 feet. What will the length of the path be?

A square garden is shown that is marked 10’ on the side. There is a path of stones along the diagonal of the square.

14.1 feet

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Brian borrowed a 20 foot extension ladder to use when he paints his house. If he sets the base of the ladder 6 feet from the house, as shown below, how far up will the top of the ladder reach?

A house is shown with a ladder leaning against it. The ladder is marked 20’, and the distance from the house to the base of the ladder is marked 6’.
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Solve Applications Using Rectangle Properties

In the following exercises, solve using rectangle properties.

The length of a rectangle is 85 feet and the width is 45 feet. What is the perimeter?

260 feet

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The length of a rectangle is 26 inches and the width is 58 inches. What is the perimeter?

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A rectangular room is 15 feet wide by 14 feet long. What is its perimeter?

58 feet

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A driveway is in the shape of a rectangle 20 feet wide by 35 feet long. What is its perimeter?

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The area of a rectangle is 414 square meters. The length is 18 meters. What is the width?

23 meters

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The area of a rectangle is 782 square centimeters. The width is 17 centimeters. What is the length?

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The width of a rectangular window is 24 inches. The area is 624 square inches. What is the length?

26 inches

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The length of a rectangular poster is 28 inches. The area is 1316 square inches. What is the width?

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Find the length of a rectangle with perimeter 124 and width 38.

24

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Find the width of a rectangle with perimeter 92 and length 19.

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Find the width of a rectangle with perimeter 16.2 and length 3.2.

4.9

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Find the length of a rectangle with perimeter 20.2 and width 7.8.

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The length of a rectangle is nine inches more than the width. The perimeter is 46 inches. Find the length and the width.

16 in., 7 in.

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The width of a rectangle is eight inches more than the length. The perimeter is 52 inches. Find the length and the width.

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The perimeter of a rectangle is 58 meters. The width of the rectangle is five meters less than the length. Find the length and the width of the rectangle.

17 m, 12 m

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The perimeter of a rectangle is 62 feet. The width is seven feet less than the length. Find the length and the width.

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The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.

13.5 m length, 12.8 m width

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The length of the rectangle is 1.1 meters less than the width. The perimeter of a rectangle is 49.4 meters. Find the dimensions of the rectangle.

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The perimeter of a rectangle is 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.

50 ft., 25 ft.

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The length of a rectangle is three times the width. The perimeter of the rectangle is 72 feet. Find the length and width of the rectangle.

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The length of a rectangle is three meters less than twice the width. The perimeter of the rectangle is 36 meters. Find the dimensions of the rectangle.

7 m width, 11 m length

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The length of a rectangle is five inches more than twice the width. The perimeter is 34 inches. Find the length and width.

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The perimeter of a rectangular field is 560 yards. The length is 40 yards more than the width. Find the length and width of the field.

160 yd., 120 yd.

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The perimeter of a rectangular atrium is 160 feet. The length is 16 feet more than the width. Find the length and width of the atrium.

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A rectangular parking lot has perimeter 250 feet. The length is five feet more than twice the width. Find the length and width of the parking lot.

85 ft., 40 ft.

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A rectangular rug has perimeter 240 inches. The length is 12 inches more than twice the width. Find the length and width of the rug.

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Everyday math

Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are six feet, eight feet and 10 feet. How many feet of fencing will she need to enclose her flowerbed?

24 feet

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Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 50 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 10 feet. How long can he make the other length?

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Writing exercises

If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.

area; answers will vary

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If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning.

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Look at the two figures below.

On the left, we have a rectangle with height 2 and width 8. On the right, we have a square with height 4 and width 4.


Which figure looks like it has the larger area?
Which looks like it has the larger perimeter?
Now calculate the area and perimeter of each figure.
Which has the larger area?
Which has the larger perimeter?

Answers will vary.
Answers will vary.
Answers will vary.
The areas are the same.
The 2x8 rectangle has a larger perimeter than the 4x4 square.

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Write a geometry word problem that relates to your life experience, then solve it and explain all your steps.

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Self check

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

This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “solve applications using triangle properties,” “use the Pythagorean Theorem,” and “solve applications using rectangle properties.” The rest of the cells are blank

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

Questions & Answers

integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
how
muqtaar
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
No; 65m/hr
albert
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
hi
albert
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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