10-1 The Pythagorean Theorem and Its Converse
Do Now
Lesson Presentation
Exit Ticket
10-1 The Pythagorean Theorem and Its Converse
Do Now #12
Classify each triangle by its angle measures.
1.
3. Simplify
2.
acute
right
12
4. If a = 6, b = 7, and c = 12, find a2 + b2 and find c2.
Which value is greater?
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
Connect to Mathematical Ideas (1)(F)
By the end of today’s lesson,
SWBAT
 Use the Pythagorean Theorem and its converse to
solve problems.
 Use Pythagorean inequalities to classify triangles.
10-1 The Pythagorean Theorem and Its Converse
Vocabulary
Pythagorean triple
10-1 The Pythagorean Theorem and Its Converse
+
a
b
b
c
c
+
a
s
+
𝑎+𝑏
c
a
c
b
=
Area
2
2
+
a
1
= 4 𝑎𝑏 + 𝑐 2
2
𝑎2 + 2𝑎𝑏 + 𝑏 2 = 2𝑎𝑏 + 𝑐 2
2
b
Area
2
𝑎 +𝑏 =𝑐
2
10-1 The Pythagorean Theorem and Its Converse
The Pythagorean Theorem
𝑎+𝑏
c
a
2
1
= 4 𝑎𝑏 + 𝑐 2
2
2
𝑎 + 2𝑎𝑏 + 𝑏 = 2𝑎𝑏 + 𝑐
2
b
2
2
𝑎 +𝑏 =𝑐
2
2
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 = x2
Simplify.
Find the positive square root.
Simplify the radical.
10-1 The Pythagorean Theorem and Its Converse
Example 2: Using the Pythagorean Theorem
Find the value of x. Give your
answer in simplest radical form.
a2 + b2 = c2
(x – 2)2 + 42 = x2
x2 – 4x + 4 + 16 = x2
–4x + 20 = 0
20 = 4x
5=x
Pythagorean Theorem
Substitute x – 2 for a, 4 for b, and x for c.
Multiply.
Combine like terms.
Add 4x to both sides.
Divide both sides by 4.
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 + b2 = c2
a2 + 262 = 362
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
a2 + b2 = c2
182 + 242 = c2
Pythagorean Theorem
c
Substitution Prop.
576 + 324 = c2
Simplify.
900 = c2
Simplify.
30 = c
Take the positive square root.
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.
c
6 ⦁3
6 ⦁4
6 ⦁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.
?
a2 + b2 = c2
132
+
842
?
= 852
?
169 + 7056 = 7225
Pythagorean Theorem.
Substitute 13 for a, 84 for b, and 85 for c.
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
Remember!
By the Triangle Inequality Theorem, the
sum of any two side lengths of a triangle
is greater than the third side length.
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?
Step 1 Determine if the measures form a triangle.
Step 2 Classify the triangle.
c2
? 2
= a + b2
Compare c2 to a2 + b2
142  62 + 112
Substitute the greatest value for c.
> 157
196 
Simplify.
Since c2 > a2 + b2, the triangle is obtuse.
10-1 The Pythagorean Theorem and Its Converse
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.
Since 5 + 8 = 13 and 13 ≯ 17, these
cannot be the side lengths of 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.
No. 162 + 482 ≠ 502
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. 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
Exit Ticket:
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.