10.1 Pythagorean Theorem
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10.1 Pythagorean Theorem I CAN: SOLVE PROBLEMS USING THE PYTHAGOREAN THEOREM IDENTIFY RIGHT TRIANGLES
Transcript of 10.1 Pythagorean Theorem
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
