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10-1 The Pythagorean Theorem and Its Converse
Do Now
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Lesson Presentation
10-1 The Pythagorean Theorem and Its Converse
Do Now #12Classify each triangle by its angle measures.
1. 2.
3. Simplify
4. If a = 6, b = 7, and c = 12, find a2 + b2 and find c2. Which value is greater?
acute right
12
85; 144; c2
10-1 The Pythagorean Theorem and Its Converse
10-1 The Pythagorean Theorem and Its Converse
How big are the screens?
10-1 The Pythagorean Theorem and Its Converse
SWBAT
Use the Pythagorean Theorem and its converse to
solve problems.
Use Pythagorean inequalities to classify triangles.
By the end of today’s lesson,
Connect to Mathematical Ideas (1)(F)
10-1 The Pythagorean Theorem and Its Converse
Pythagorean triple
Vocabulary
10-1 The Pythagorean Theorem and Its Converse
a
b
c
a
bc a
b
c
a
bc
=
𝑎 + 𝑏 2
s 2
= 41
2𝑎𝑏 + 𝑐2
+
+
+
+
Area Area
𝑎2 + 2𝑎𝑏 + 𝑏2 = 2𝑎𝑏 + 𝑐2
𝑎2 + 𝑏2 = 𝑐2
10-1 The Pythagorean Theorem and Its Converse
a
b
c
𝑎 + 𝑏 2 = 41
2𝑎𝑏 + 𝑐2
𝑎2 + 2𝑎𝑏 + 𝑏2 = 2𝑎𝑏 + 𝑐2
𝑎2 + 𝑏2 = 𝑐2
The Pythagorean Theorem
10-1 The Pythagorean Theorem and Its Converse
10-1 The Pythagorean Theorem and Its Converse
Example 1: Using the Pythagorean Theorem
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 = x2Simplify.
Find the positive square root.
Simplify the radical.
10-1 The Pythagorean Theorem and Its Converse
Pythagorean Theorem
(x – 2)2 + 42 = x2 Substitute x – 2 for a, 4 for b, and x for c.
x2 – 4x + 4 + 16 = x2 Multiply.
–4x + 20 = 0 Combine like terms.
20 = 4x Add 4x to both sides.
5 = x Divide both sides by 4.
Example 2: Using the Pythagorean Theorem
Find the value of x. Give your answer in simplest radical form.
a2 + b2 = c2
10-1 The Pythagorean Theorem and Its Converse
Example 3: Find the Distance
Dog agility courses often contain a seesaw obstacle, as shown below. To the nearest inch, how far above the ground are the dog’s paws when the seesaw is parallel to the ground?
a2 + 262 = 362
a2 + b2 = c2
a2 + 676 = 1296
a2 = 620
a ≈ 24.8997992
∴ The dog’s paws are 25 in. above the ground.
10-1 The Pythagorean Theorem and Its Converse
10-1 The Pythagorean Theorem and Its Converse
Example 4: Identifying Pythagorean Triples
Find the value of c. Give your answer in simplest radical form.
Method 1 Pythagorean Theorema2 + b2 = c2
182 + 242 = c2Substitution Prop.
576 + 324 = c2 Simplify.
900 = c2 Simplify.
30 = c Take the positive square root.
c
10-1 The Pythagorean Theorem and Its Converse
Example 4: Identifying Pythagorean Triples
Find the value of c. Give your answer in simplest radical form.
Method 2
Sometimes you can use a Pythagorean triple and mental math to find the length of a side of a right triangle. 6 ⦁ 3
6 ⦁ 4
c6 ⦁ 5
10-1 The Pythagorean Theorem and Its Converse
Example 4: Identifying a Right Triangle
A triangle has side lengths 85, 84, and 13. Is the triangle a right triangle? Explain.
Pythagorean Theorem.a2 + b2 = c2?
132 + 842 = 852 Substitute 13 for a, 84 for b, and 85 for c.?
169 + 7056 = 7225 Simplify.?
7225 = 7225
∴ Yes, it’s a right ∆ because 132 + 842 = 852
10-1 The Pythagorean Theorem and Its Converse
10-1 The Pythagorean Theorem and Its Converse
To understand why the Pythagorean inequalities are true, consider ∆ABC.
10-1 The Pythagorean Theorem and Its Converse
By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater than the third side length.
Remember!
10-1 The Pythagorean Theorem and Its Converse
Example 5: Classifying Triangle
A triangle has side lengths 6, 11, and 14. Is it an acute, obtuse, or a right triangle?
Compare c2 to a2 + b2
Substitute the greatest value for c.
Simplify.
142 62 + 112
196 157>
Since c2 > a2 + b2, the triangle is obtuse.
c2 = a2 + b2?
Step 1 Determine if the measures form a triangle.
Step 2 Classify the triangle.
10-1 The Pythagorean Theorem and Its Converse
Since 5 + 8 = 13 and 13 ≯ 17, these cannot be the side lengths of a triangle.
Example 5: Classifying Triangle
A triangle has side lengths 5, 8, and 17. Is it an acute, obtuse, or a right triangle?
Step 1 Determine if the measures form a triangle.
10-1 The Pythagorean Theorem and Its Converse
Got It ? Solve With Your Partner
Problem 1 Finding the Length of the hypotenuse.
The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse ?
26
10-1 The Pythagorean Theorem and Its Converse
Got It ? Solve With Your Partner
Problem 2 Finding the Length of the hypotenuse.
The size of a computer monitor is the length of its diagonal. You want to buy a 19-in. monitor that has a height of 11 in. What is the width of the monitor ? Round to the nearest tenth of an inch.
15.5 in.
10-1 The Pythagorean Theorem and Its Converse
Got It ? Solve With Your Partner
Problem 3 Identifying a Right Triangle
a. A triangle has side lengths 16, 48, and 50. Is the triangle a right triangle? Explain your reasoning.
b. Once you know which length represents the hypotenuse, does it matter which length you substitute for a and which length you substitute for b? Explain.
No. 162 + 482 ≠ 502
No. a2 + b2 = b2 + a2
10-1 The Pythagorean Theorem and Its Converse
Got It ? Solve With Your Partner
Problem 4 Classifying a Triangle
Is a triangle with side lengths 7, 8, and 9 an acute, obtuse, or a right? Explain.
acute
10-1 The Pythagorean Theorem and Its Converse
Closure: Communicate Mathematical Ideas (1)(G)
What is the difference between the ways the Pythagorean Theorem and its converse are used?
The Pythagorean Theorem is used to determine the
length of the third side of a right triangle given two of
the sides. The converse is used to determine whether
three given side lengths form a right triangle.
10-1 The Pythagorean Theorem and Its Converse
1. Find the value of x.
2. An entertainment center is 52 in. wide and 40 in. high. Will a TV with a 60 in. diagonal fit in it? Explain.
3. 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.
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