Pythagorean Theorem 5.4. Learn the Pythagorean Theorem. Define Pythagorean triple. Learn the...

Pythagorean Theorem 5.4

• Learn the Pythagorean Theorem.

• Define Pythagorean triple.

• Learn the Pythagorean Inequality.

• Solve problems with the Pythagorean Theorem and Inequality.

The Pythagorean Theorem is probably the most famous mathematical relationship. It states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

a2 + b2 = c2

Find the value of x. Give your answer in simplest radical form.

a2 + b2 = c2 Pythagorean Theorem

22 + 62 = x2 Substitute 2 for a, 6 for b, and x for c.

40 = x2 Simplify.

Find the positive square root.

Simplify the radical.

Find the value of x. Give your answer in simplest radical form.

a2 + b2 = c2 Pythagorean Theorem

42 + 82 = x2 Substitute 4 for a, 8 for b, and x for c.

80 = x2 Simplify.

Find the positive square root.

Simplify the radical.

A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a Pythagorean triple.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

The side lengths are nonzero whole numbers that satisfy the equation a2 + b2 = c2, so they form a Pythagorean triple.

a2 + b2 = c2 Pythagorean Theorem

142 + 482 = c2 Substitute 14 for a and 48 for b.

2500 = c2 Multiply and add.

50 = c Find the positive square root.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

a2 + b2 = c2 Pythagorean Theorem

82 + 102 = c2

164 = c2

The side lengths do not form a Pythagorean triple because is not a whole number.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

The side lengths are nonzero whole numbers that satisfy the equation a2 + b2 = c2, so they form a Pythagorean triple.

a2 + b2 = c2 Pythagorean Theorem

242 + b2 = 262 Substitute 24 for a and 26 for c.

b2 = 100 Multiply and subtract.

b = 10 Find the positive square root.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

No. The side length 2.4 is not a whole number.

The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths.

You can also use side lengths to classify a triangle as acute or obtuse.

A

B

C

c

b

a

To understand why the Pythagorean inequalities are true, consider ∆ABC.

The third side of a triangle is less than the sum and greater than the difference of the other two sides.

Remember!

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 7, 10

Since 2< 10 < 12; 5, 7, and 10 can be the side lengths of a triangle.

Classify the triangle.

Since c2 > a2 + b2, the triangle is obtuse.

Add and compare.100 > 74

Multiply.

Compare c2 to a2 + b2.

Substitute the longest side for c.

c2 = a2 + b2?

102 = 52 + 72?

100 = 25 + 49?

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

Determine if the measures form a triangle.

5, 8, 17

Since 3 < 17 < 13, these cannot be the side lengths of a triangle.

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 7, 12, 16

Since 5 < 16 < 19; 7, 12, and 16 can be the side lengths of a triangle.

Classify the triangle.

Since c2 > a2 + b2, the triangle is obtuse.

Add and compare.256 > 193

Multiply.

Compare c2 to a2 + b2.

Substitute the longest side for c.

c2 = a2 + b2?

162 = 122 + 72?

256 = 144 + 49?

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

Determine if the measures form a triangle.

11, 18, 34

Since 7 < 34 < 29 ; these cannot be the sides of a triangle.

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 3.8, 4.1, 5.2

Since .3< 5.2 < 7.9; 3.8, 4.1, and 5.2 can be the side lengths of a triangle.

Classify the triangle.

Since c2 < a2 + b2, the triangle is acute.

Add and compare.27.04 < 31.25

Multiply.

Compare c2 to a2 + b2.c2 = a2 + b2?

5.22 = 3.82 + 4.12?

27.04 = 14.44 + 16.81 ?

Substitute the longest side for c.

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

13; yes; the side lengths are nonzero whole numbers that satisfy Pythagorean’s Theorem.

Tell if the measures 7, 11, and 15 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

yes; obtuse

225b25 2

222 cba 222 15b5

feet1.14b210b200b 200b2

288x2

222 24xx 576x2 2

meters0.17x212x288x

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

a.a.

b.b.

Yes a triangle can be formed 1 < 6 < 9Yes a triangle can be formed 1 < 6 < 9

4136

1625_36

45_6 222

AcuteAcute

Yes a triangle can be formed 16 < 26 < 36Yes a triangle can be formed 16 < 26 < 36

676676

576100_676

2410_26 222

RightRight

c.c. Yes a triangle can be formed 5 < 25 < 35Yes a triangle can be formed 5 < 25 < 35

625625

400225_625

2015_25 222

RightRight

Find the exact answer of the missing side.Find the exact answer of the missing side.

a.a. b.b. c.c. 20x

d.d. e.e. f.f.

23x18x

10x

52x20x

35x75x 55x125x

a.a. b.b. c.c.x = 25x = 25 x = 34x = 34 17x

d.d. e.e. f.f.

Find the exact answer of the missing side.Find the exact answer of the missing side.

113x99x 33x 105x

a.a. b.b. c.c.x = 15x = 15 x = 51x = 51 39x

d.d. e.e. f.f.x = 60x = 60 x = 30x = 30

Find the exact answer of the missing side.Find the exact answer of the missing side.

33x27x

a.a. b.b. c.c.x = 15x = 15 x = 51x = 51

d.d. e.e. f.f.

Find the exact answer of the missing side.Find the exact answer of the missing side.

39x

616x

1542x

x = 82x = 82

1172x

468x

AssignmentAssignment

Section 5.4 Section 5.4 8 - 308 - 30