2. The Pythagorean Theorem describes the relationship between the length of the hypotenuse c and the lengths of the legs a & b of a right triangle.
In a right triangle the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. a2
+ b2
= c2 a b c

3. The sides a and b adjacent to the right angle are the legs of the right triangle.
The side c of the right triangle is the hypotenuse of the right triangle. c a b

4. a2 + b2 = c2 62 + 82 = c2 36 + 64 = c2 100 = c2 10 = c 8 in. = b c= x
You can use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle if the measures of both legs are known.
a2 + b2 = c2
= c2
8 in.
= b c= x
= c2 
100 = c2 
6 in.
= a
10 = c
The length of the hypotenuse of this right triangle is 10 in.

5. a2 + b2 = c2 =c =a =b 52 132 + b2 = 25 + b2 = 169 b2 = 144 b = 12 −25
You can also use the Pythagorean Theorem to find the measure of a leg of a right triangle if the measure of the other leg and the hypotenuse are known.
a2 + b2 = c2 =c
13 cm 52
132
+ b2 = =a
5 cm
25 + b2 = 169
−25
−25 =b X 
b2 = 144 
b = 12
The missing measure of the leg of this right triangle is 12 cm.

6. B A c= 15m a= 9m c= x b= 4 in. b= x a= 3 in. a2 + b2 = c2 a2 + b2 = c2
Use the Pythagorean Theorem to find the length of the missing sides:
Try These: B A c=
15m a= 9m c= x b=
4 in. b= x a=
3 in.
a2 + b2 = c2
a2 + b2 = c2
92 + x2 = 152
= x2
81 + x2 = 225
= x2
−81
−81 
25 = x2 
x2 = 144  
5 in. = x
x = 12 m

7. Use the Pythagorean Theorem to determine whether the triangle with these side lengths is a right triangle: 5 ft., 7ft., 8ft
The hypotenuse is the longest side, so c = 8. The smaller numbers will be either of the two legs, so we will let a = 5 & b = 7.
a2 + b2 = c2
= 82
= 64
74 = 64
The triangle is not a right triangle.

8. Try This:
Use the Pythagorean Theorem to determine whether the triangle with these side lengths is a right triangle: 10 ft., 26ft., 24ft
The hypotenuse is the longest side, so c = 26. The smaller numbers will be either of the two legs, so we will let a = 10 & b = 24.
a2 + b2 = c2
= 262
= 676
676 = 676
The triangle is a right triangle.

9. a2 + b2 = c2 122 + 162 = x2 c= a= 144 + 256 = x2  400 = x2  b=
A boat sailed due west 16 miles and then due north 12 miles. How far is the boat, in a straight line, from its starting point?
Make a drawing and use the Pythagorean Theorem to solve this problem.
a2 + b2 = c2
north
= x2 c= a=
x mi
= x2
12 mi 
= x2 
west
start b=
16 mi
20 = x
The boat is 20 miles from its starting point.

10. a2 + b2 = c2 =c 82 + 42 = x2 64 + 16 = x2 80 = x2 x ≈ 8.9 a=   =b
A gardener marked off a rectangular planting site 4 meters wide and 8 meters long. What would be the measure of the diagonal of the planting site? Round your answer to the nearest tenths.
Hint: Make a drawing and use the Pythagorean Theorem to solve this problem.
4 m
X m
8 m
a2 + b2 = c2 =c a=
= x2
= x2
80 = x2   =b
x ≈ 8.9
The length of the diagonal is about 8.9 meters.