The Pythagorean Theorem: Detailed Explanation

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NEO Fri, Feb 04, 2022 10:37 AM UTC

This pdf covers the basics of the Pythagoras theorem and how to find an unknown side by using this theorem with examples.

1. Section 2.1: The Pythagorean Theorem
The Pythagorean Theorem is a formula that gives a relationship between the sides of a right
triangle The Pythagorean Theorem only applies to RIGHT triangles. A RIGHT triangle is a
triangle with a 90 degree angle. The 90 degree angle in a right triangle is often depicted with a
a c Pythagorean Theorem: a2 + b2 = c2
b
NOTE: The side “c” is always the side opposite the right angle. Side “c” is called the
The sides adjacent to the right angle are named “a” and “b”.
There is no choice when selecting a side to be named “c”, but the sides you choose to call “a”
and “b” are interchangeable.
Example: Name each side of the triangle below. That is name one side a, one b and the
hypotenuse c.
The c is the side opposite the right angle. There is no flexibility when it comes to choosing the
side we name c.
The sides a, b are adjacent to the right angle and we have flexibility when naming them.
Each of these is a correct answer.
Answer 1: a = 3, b = 4 and c = 5
Answer 2: a = 4, b = 3, and c = 5

2. Example: Find the length of the missing side. Round to 2 decimals. Make sure to use the
proper units in your answer.
Step 1: Name each side. (It isn’t necessary to use the units (cm) until I write my answer.)
The side with the question mark is opposite the right angle. It needs to be called c.
The 2 and the 4 are the sides that need to be labeled a and b. I have a bit of flexibility.
It would be correct to call:
a = 2 and b = 4 or a = 4 and b = 2
I will pick a = 2 and b = 4
Step 2: put the values into the a2 + b2 = c2 formula and do the algebra to solve for the missing
22 + 42 = c2
4 + 16 = c2
20 = c2
√20 = c
Answer: c = 4.47 cm
Example: Find the length of the missing side. Make sure to use proper units in your answer.
Step 1: Name each side. (It isn’t necessary to use the units (cm) until I write my answer.)
The side opposite the right angle is called c. c = 20
The side with the x and the 12 are the “a” and “b” for this problem. I will call a = 12 and
b=x
Step 2: Put the values into the a2 + b2 = c2 formula and do the algebra to solve for the missing
122 + x2 = 202
144 + x2 = 400
-144 -144
x2 = 256
x = √256
Answer: x = 16 cm

3. Example: Find the length of the missing side (round your answer to 2 decimals). Be sure to
include units in your answer.
Step 1: Name each side.
The side opposite the right angle is called c. c = 6.2
The x and the 4.9 are the sides for “a” and “b” in this problem. I will call a = 4.9 and
b=x
Step 2: put the values into the a2 + b2 = c2 formula and do the algebra to solve for the missing
4.92 + x2 = 6.22
24.01 + x2 = 38.44
-24.01 -24.01
x2 = 14.43
𝑥 = √14.43
x = 3.798
Answer: x = 3.80 m

4. Example: Find x. Be sure to include units in your answer.
This is a Pythagorean Theorem problem in disguise. I can break this rectangle up into these two
Now I just need to solve for x. Each of the triangles has the same three sides and will give the
same answer regardless of the triangle I use when I solve for x.
Step 1: Name each side. I will use the left triangle.
The side opposite the right angle is called c. c = 10
The x and the 8 are the “a” and “b” for this problem. I will call a = 8 and b = x
Step 2: put the values into the a2 + b2 = c2 formula and do the algebra to solve for the missing
82 + x2 = 102
64 + x2 = 100
-64 -64
x2 = 36
𝑥 = √36
Answer: x = 6 cm (I put my units back in at the end. I don’t do my computations with the units
in place as they don’t affect my algebra.)

5. Example: Find the height of the triangle. That is, solve for h (round your answer to 2 decimal
places.) Make sure to put units with your answer.
This is also a Pythagorean Theorem problem in disguise. I can break this triangle up into these
two triangles.
Now I just need to solve for h. I will use triangle on the right as it has all three sides labeled.
Step 1: Name each side.
The side opposite the right angle is called c.
The h and the 4 are the “a” and “b” for this problem. I will call a = 4 and b = h
Step 2: Put the values into the a2 + b2 = c2 formula and do the algebra to solve for the missing
42 + h2 = 82
16 + h2 = 64
-16 -16
h2 = 48
ℎ = √48
h = 6.928
Answer: h = 6.93 ft (I put units in my answer as each number in the problem has units.)

6. Homework #1-20: Use the Pythagorean Theorem to find the missing length. Round to two
decimal places when necessary. Make sure to use units in your answer when the lengths are
given with units.
1) 2)
3) 4)
5) Round your answer to 2 decimals. 6)

7. 7) Find the length of the diagonal. Round 8) Find the length of the diagonal. Round to
your answer to 2 decimals. 2 decimals.
9) Find the length of the diagonal. 10) Find the length of the diagonal.
11) Solve for x. Assume the units are in 12) Find the height of the triangle. Assume
meters. Round to 2 decimals. the lengths are given in meters.

8. 13) Find d and D. Round to 2 decimals. Hint 14) Find the lengths of the dashed lines.
the “ ‘ “ means the units are in feet. Assume sides are measured in feet.
15) Solve for c and d. 16) Find d.
1) 13 cm 3) x = 45 cm 5) a = 8.94 in 7) 14.14 m 9) 40.25 m 11) 10.91 m
13) d = 12.73’ D = 15.59’ 15) c = 10 in d = 14.14 in

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