Learn to use the Pythagorean Theorem to find the length of a side of a right triangle.

1
Learn to use the Pythagorean Theorem to find the
length of a side of a right triangle.
2
Insert Lesson Title Here
Vocabulary
leg hypotenuse Pythagorean Theorem
3
In a right triangle, the two sides that form the
right angle are called legs. The side opposite
the right angle is called the hypotenuse.
Hypotenuse
Leg
Leg
One of the first people to recognize the
relationship between the sides of a right
triangle was the Greek mathematician Pythagoras.
This special relationship is called the
Pythagorean Theorem.
4


PYTHAGOREAN THEOREM
In a right triangle, the sum of the squares of
the lengths of the legs is equal to the square of
the length of the hypotenuse.
c
a
a2 b2 c2
b
You can use the Pythagorean Theorem to find the
length of any side of a right triangle.
5
Additional Example 1A Calculating the Length of
a Side of a Right Triangle
Use the Pythagorean Theorem to find the missing
measure.
c
12 cm
16 cm
Use the Pythagorean Theorem.
a2 b2 c2
Substitute for a and b.
122 162 c2
Evaluate the powers.
144 256 c2
Add.
400 c2
Take the square root of both sides.
20 c
The length of the hypotenuse is 20 cm.
6
Additional Example 1B Calculating the Length of
a Missing Side of a Right Triangle
Use the Pythagorean Theorem to find the missing
measure.
b
5 cm
13 cm
Use the Pythagorean Theorem.
a2 b2 c2
Substitute for a and c.
52 b2 132
25 b2 169
Evaluate the powers.
25
25
Subtract 25 from each side.
b2 144
Take the square root of both sides.
b 12
The length of the missing leg is 12 cm.
7
Check It Out Example 1A
Use the Pythagorean Theorem to find the missing
measure.
c
11 cm
15 cm
Use the Pythagorean Theorem.
a2 b2 c2
Substitute for a and b.
112 152 c2
Evaluate the powers.
121 225 c2
Add.
346 c2
Take the square root of both sides.
18.6 ? c
The length of the hypotenuse is about 18.6 cm.
8
Check It Out Example 1B
Use the Pythagorean Theorem to find the missing
measure.
b
3 cm
5 cm
Use the Pythagorean Theorem.
a2 b2 c2
Substitute for a and c.
32 b2 52
9 b2 25
Evaluate the powers.
9
9
Subtract 9 from each side.
b2 16
Take the square root of both sides.
b 4
The length of the missing leg is 4 cm.
9
Additional Example 2 Problem Solving Application
A square field has sides of 75 feet. About how
far is it from one corner of the field to the
opposite corner of the field? Round your answer
to the nearest tenth.
10
Additional Example 2 Continued
Rewrite the question as a statement.
Find the distance from one corner of the field
to the opposite corner of the field.
List the important information
Drawing a segment from one corner of the field
to the opposite corner of the field divides the
field into two right triangles.
The segment between the two corners is the
hypotenuse.
The sides of the field are legs, and they
are each 75 feet long.
11
Additional Example 2 Continued
You can use the Pythagorean Theorem to write an
equation.
12
Additional Example 2 Continued
a2 b2 c2
Use the Pythagorean Theorem.
Substitute for the known variables.
752 752 c2
5,625 5,625 c2
Evaluate the powers.
11,250 c2
Add.
Take the square roots of both sides.
106.066012 ? c
Round.
106.1 ? c
The distance from one corner of the field to the
opposite corner is about 106.1 feet
13
Additional Example 2 Continued
Look Back
The hypotenuse is the longest side of a right
triangle. Since the distance from one corner of
the field to the opposite corner is greater than
the length of a side of the field, the answer is
reasonable.
14
Insert Lesson Title Here
Check It Out Example 2
A rectangular field has a length of 100 yards and
a width of 33 yards. About how far is it from one
corner of the field to the opposite corner of the
field? Round your answer to the nearest tenth.
Rewrite the question as a statement.
Find the distance from one corner of the field
to the opposite corner of the field.
15
Check It Out Example 2 Continued
List the important information
Drawing a segment from one corner of the field
to the opposite corner of the field divides the
field into two right triangles.
The segment between the two corners is the
hypotenuse.
The sides of the fields are legs, and they are
33 yards long and 100 yards long.
You can use the Pythagorean Theorem to write an
equation.
16
Insert Lesson Title Here
Check It Out Example 2 Continued
a2 b2 c2
Use the Pythagorean Theorem.
332 1002 c2
Substitute for the known variables.
1089 10,000 c2
Evaluate the powers.
11,089 c2
Add.
105.3043208 ? c
Take the square roots of both sides.
105.3 ? c
Round.
The distance from one corner of the field to the
opposite corner is about 105.3 yards.
17
Facts about Sides and angles

• How to determine if a triangle is a right
  triangle.
• The largest square is less than the smaller
  squares combined, . Acute triangle
• The largest square is equal to the smaller
  squares combined, . Right triangle
• The largest square is greater than the smaller
  square combined, . Obtuse triangle

a2 b2 gt c2
a2 b2 c2
a2 b2 lt c2