10.1 Pythagorean TheoremI CAN:

-SOLVE PROBLEMS USING THE PYTHAGOREAN THEOREM

-IDENTIFY RIGHT TRIANGLES

Solve It!

Pearson Warm Up

Flocab!

http://www.flocabulary.com/pythagorean-theorem/

BIG Ideas

The lengths of the sides of a right triangle have a special relationship.

BIG Ideas

The lengths of the sides of a right triangle have a special relationship.

If the lengths of any two sides of a right triangle are known, the length of the third side can be found.

BIG Ideas

The lengths of the sides of a right triangle have a special relationship.

If the lengths of any two sides of a right triangle are known, the length of the third side can be found.

If the length of each side of a triangle is known, then whether the triangle is a right triangle can be determined.

FOCUS QUESTION: Why is the Pythagorean Theorem useful?

Vocabulary to Know

Hypotenuse The side opposite the right angle

Vocabulary to Know

Leg Each of the sides forming the right angle

Vocabulary to Know

Pythagorean Theorem In any right triangle, the sum of the squares of the lengths of the legs

is equal to the square of the length of the hypotenuse.

a2 + b2 = c2

Finding the Hypotenuse and Leg Lengths

You can use the Pythagorean Theorem to find the length of a right triangle’s hypotenuse given the lengths of its legs.

Using the Pythagorean Theorem to solve for a side length involves finding a principal square root because side lengths are always positive.

Finding the Hypotenuse

The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown?

6 in.

6 in.

Finding the Hypotenuse

The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown?

Use the Pythagorean Theorem a2 + b2 = c2

6 in.

6 in.

Finding the Hypotenuse

The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown?

Use the Pythagorean Theorem a2 + b2 = c2

Substitute 6 for a and b. 62 + 62 = c2

6 in.

6 in.

Finding the Hypotenuse

The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown?

Use the Pythagorean Theorem a2 + b2 = c2

Substitute 6 for a and b. 62 + 62 = c2

Simplify 36 + 36 = c2

6 in.

6 in.

Finding the Hypotenuse

The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the right triangle shown?

Use the Pythagorean Theorem a2 + b2 = c2

Substitute 6 for a and b. 62 + 62 = c2

Simplify 36 + 36 = c2

Simplify 72 = c2

6 in.

6 in.

Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the

hypotenuse of the right triangle shown?

Use the Pythagorean Theorem a2 + b2 = c2

Substitute 6 for a and b. 62 + 62 = c2

Simplify 36 + 36 = c2

Simplify 72 = c2

Find the principal square root. = c

6 in.

6 in.

Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the

hypotenuse of the right triangle shown?

Use the Pythagorean Theorem a2 + b2 = c2

Substitute 6 for a and b. 62 + 62 = c2

Simplify 36 + 36 = c2

Simplify 72 = c2

Find the principal square root. = c

Use a calculator 8.5 ≈ c

Convert to a fraction!!

6 in.

6 in.

Finding the Hypotenuse The tiles are squares with 6-in. sides. What is the length of the hypotenuse

of the right triangle shown?

Use the Pythagorean Theorem a2 + b2 = c2

Substitute 6 for a and b. 62 + 62 = c2

Simplify 36 + 36 = c2

Simplify 72 = c2

Find the principal square root. = c

Use a calculator 8.5 ≈ c

Convert to a fraction!!

Write answer sentence.

The hypotenuse is 8 ½ in.

6 in.

6 in.

Finding the hypotenuse – Dry erase practice

What is the length of the hypotenuse of a right triangle with leg lengths 9 cm and 12 cm?

Finding the Length of a Leg

What is the side length b in the triangle?

13 cm

b

5 cm

Finding the Length of a Leg What is the side length b in the triangle?

Use the Pythagorean Theorem. a2 + b2 = c2

13 cm

b

5 cm

Finding the Length of a Leg What is the side length b in the triangle?

Use the Pythagorean Theorem. a2 + b2 = c2

Substitute 5 for a and 13 for c. 52 + b2 = 132

13 cm

b

5 cm

Finding the Length of a Leg What is the side length b in the triangle?

Use the Pythagorean Theorem. a2 + b2 = c2

Substitute 5 for a and 13 for c. 52 + b2 = 132

Simplify. 25 + b2 = 169

13 cm

b

5 cm

Finding the Length of a Leg What is the side length b in the triangle?

Use the Pythagorean Theorem. a2 + b2 = c2

Substitute 5 for a and 13 for c. 52 + b2 = 132

Simplify. 25 + b2 = 169

Remember how we solve equations. We need to isolate the

variable by using the inverse operation.

13 cm

b

5 cm

Finding the Length of a Leg What is the side length b in the triangle?

Use the Pythagorean Theorem. a2 + b2 = c2

Substitute 5 for a and 13 for c. 52 + b2 = 132

Simplify. 25 + b2 = 169

Remember how we solve equations. We need to isolate the

variable by using the inverse operation.

Subtract 25 from each side. b2 = 144

13 cm

b

5 cm

Finding the Length of a Leg What is the side length b in the triangle?

Use the Pythagorean Theorem. a2 + b2 = c2

Substitute 5 for a and 13 for c. 52 + b2 = 132

Simplify. 25 + b2 = 169

Remember how we solve equations. We need to isolate the

variable by using the inverse operation.

Subtract 25 from each side. b2 = 144

Find the principal square root of each side. b = 72

13 cm

b

5 cm

Find the Length of a Leg – Dry Erase Practice

What is the side length a in the triangle?

15

12

a

Vocabulary to Know

Conditional An if-then statement such as “If an animal is a horse, then it has four

legs.”

Vocabulary to Know

Hypothesis The “if” part of the conditional.

Conclusion The “then” part of the conditional.

Converse Switches the hypothesis and conclusion.

Conditional

You can write the Pythagorean Theorem as a conditional: “If a triangle is a right triangle with legs of lengths a and b and

hypotenuse of length c, then a2 + b2 = c2.

The converse of the Pythagorean Theorem is always true.

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

6 in., 24 in., 25 in.

4 m, 8 m, 10 m

10 in., 24 in., 26 in.

8 ft., 15 ft., 16 ft.

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

6 in., 24 in., 25 in.

62 + 242 ___ 252

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

6 in., 24 in., 25 in.

62 + 242 ___ 252

36 + 576 ___ 625

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

6 in., 24 in., 25 in.

62 + 242 ___ 252

36 + 576 ___ 625

612 ≠ 625

NO

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

4 m, 8 m, 10 m

42 + 82 = 102

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

4 m, 8 m, 10 m

42 + 82 ___ 102

16 + 64 ___ 100

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

4 m, 8 m, 10 m

42 + 82 ___ 102

16 + 64 ___ 100

80 ≠ 100

NO

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

10 in., 24 in., 26 in.

102 + 242 ___ 262

100 + 576 ___ 676

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

10 in., 24 in., 26 in.

102 + 242 ___ 262

100 + 576 ___ 676

676 = 676

YES

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

8 ft., 15 ft., 16 ft.

82 + 152 ___ 162

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

8 ft., 15 ft., 16 ft.

82 + 152 ___ 162

64 + 225 ___ 256

Identifying Right Triangles

Which set of lengths could be the side lengths of a right triangle?

8 ft., 15 ft., 16 ft.

82 + 152 ___ 162

64 + 225 ___ 256

289 ≠ 256

Identifying Right TrianglesDry Erase Practice

Could the lengths 20 mm, 47 mm, and 52 mm be the side lengths of a right triangle? Explain.

Focus Question Answer

Why is the Pythagorean Theorem useful?

BIG Ideas

The lengths of the sides of a right triangle have a special relationship.

If the lengths of any two sides of a right triangle are known, the length of the third side can be found.

If the length of each side of a triangle is known, then whether the triangle is a right triangle can be determined.

Assignment

Pages 628-630

1-3

6-19

20-28 even

29-40