Applications Using Square Roots
This section will discuss applications which use square roots, in
particular the Pythagorean Theorem. As always, the following steps will
help to translate and solve the problem.
1.
2.
3.
4.
5.
Read through the entire problem
Organize the information (a picture may be useful)
Write the equation
Solve the equation
Check the answer
Pythagorean Theorem
If we are given the measurements of two sides of a right triangle, we can
easily find the measurement of the third side by using the Pythagorean
Theorem.
The Pythagorean Theorem states that the square of
hypotenuse is equal to the sum of the squares of the other two sides.
The formula is:
a2 + b2 = c2
where c is the hypotenuse; a and b are the legs
Example 1:
Find the length of the hypotenuse of the right triangle pictured below:
7
24
c2 = a2 + b2
Use the Pythagorean Theorem
2
2
2
c = 7 + 24
Substitute the values into the formula
2
c = 49 + 576 Simplify the squares
c2 = 625
Add
√c 2 = √625
c = 25
Take the square root of each side
To check the answer: 252 = 72 + 242
625 = 49 + 576
625 = 625 True
Example 2:
Find the missing length of the leg of the right triangle pictured below:
13
5
a2 + b2 = c2
a2 + 52 = 132
Use the Pythagorean Theorem
Substitute the values into the formula
a2 + 25 = 169
- 25 - 25
2
a
= 144
Simplify the squares
Subtract 25 from both sides
√a2 = √144
a = 12
Take the square root of each side
To check the answer:
122 + 52 = 132
144 + 25 = 169
169 = 169
True
Example 3:
Given a rectangular picture frame that is 12 inches by 10 inches, how long is the
diagonal from one corner to the other. Round your answer to the nearest tenth.
Draw and label a picture
10 inches
12 inches
c2 = a2 + b2
c2 = 122 + 102
c2 = 144 + 100
c2 = 244
Use the Pythagorean Theorem
Substitute the values into the formula
Simplify the squares
Add
√c 2 = √244
Take the square root of each side
c ≈ 15.6204.. Round the answer to the nearest 10th
c ≈ 15.6 in
Example 4:
A 6 foot ladder is placed against a wall. If the base of the ladder is 3 feet from the
the wall, how high up the wall will the ladder reach? Round your answer to the
nearest hundredth.
Draw and label a picture
6ft ladder
Wall
3 ft
2
2
2
a +b =c
a2 + 32 = 62
a2 + 9 = 36
-9 -9
2
a
= 27
Use the Pythagorean Theorem
Substitute the values into the formula
Simplify the squares
Subtract 9 from both sides
√a2 = √27
Take the square root of each side
a ≈ 5.1961.. Round the answer to the nearest hundredth
a ≈ 5.20 ft