1. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Check Skills You’ll Need
(For help, go to the Skills Handbook, page 753.)
Square the lengths of the sides of each triangle. What do you notice?
Check Skills You’ll Need
8-1

2. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Check Skills You’ll Need
Solutions
1. 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; = 25
2. 52 = (5)(5) = 25; 122 = (12)(12) = 144; 132 = (13)(13) = 169;
= 169
3. 62 = (6)(6) = 36; 82 = (8)(8) = 64; 102 = (10)(10) = 100; = 100
4. 42 = (4)(4) = 16; (4 2)2 = (4 2)(4 2) = (2)(2) = 16(2) = 32; = 32.
8-1

3. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Notes
8-1

4. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Notes
A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2.
Some common Pythagorean triples are:
3, 4, , 12, , 15, , 24, 25
8-1

5. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Notes
8-1

6. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Notes
8-1

7. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Notes
8-1

8. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Additional Examples
Pythagorean Triples
A right triangle has legs of length 16 and 30. Find the length of the hypotenuse. Do the lengths of the sides form a Pythagorean triple?
a2 + b2 = c2 Use the Pythagorean Theorem.
= c2 Substitute 16 for a and 30 for b.
= c2 Simplify.
1156 = c2
34 = c Take the square root.
The length of the hypotenuse is 34.
The lengths of the sides, 16, 30, and 34, form a Pythagorean triple because they are whole numbers that satisfy a2 + b2 = c2. Notice
that each length is twice the common Pythagorean triple of 8, 15,
and 17.
Quick Check
8-1

9. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Additional Examples
Using Simplest Radical Form
Find the value of x. Leave your answer in simplest radical form.
a2 + b2 = c2 Use the Pythagorean Theorem.
x = 122 Substitute x for a, 10 for b, and 12 for c.
x = Simplify.
x2 = 44 Subtract 100 from each side.
x = 4(11) Take the square root of each side.
x = Simplify.
Quick Check
8-1

10. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Additional Examples
Real-World Connection
A baseball diamond is a square with 90-ft sides. Home plate and second base are at opposite vertices of the square. About how far is home plate from second base?
Use the information to draw a baseball diamond.
a2 + b2 = c Use the Pythagorean Theorem.
= c Substitute 90 for a and for b.
8, ,100 = c2 Simplify.
16,200 = c2
c = 16, Take the square root.
c Use a calculator.
The distance to home plate from second base is about 127 ft.
Quick Check
8-1

11. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Additional Examples
Is It a Right Triangle?
Is this triangle a right triangle?
a2 + b2 c2
Substitute 4 for a, 6 for b, and 7 for c.
Simplify.
52 ≠ 49
Because a2 + b2 ≠ c2, the triangle is not a right triangle.
Quick Check
8-1

12. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Additional Examples
Classifying Triangles as Obtuse, Acute, or Right
The numbers represent the lengths of the sides of a
triangle. Classify each triangle as acute, obtuse, or right.
a.15, 20, 25
c2 a2 + b Compare c2 with a2 + b2.
Substitute the greatest length for c.
Simplify.
625 = 625
Because c2 = a2 + b2, the triangle is a right triangle.
8-1

13. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Additional Examples
(Continued)
b. 10, 15, 20
c a2 + b Compare c 2 with a2 + b 2.
Substitute the greatest length for c.
Simplify.
Because c a2 + b2, the triangle is obtuse.
Quick Check
8-1

14. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
Lesson 8-1
Lesson Quiz
1. Find the value of x.
2. Find the value of x. Leave your
answer in simplest radical form.
3. The town of Elena is 24 mi north and 8 mi
west of Holberg. A train runs on a straight
track between the two towns. How many
miles does it cover? Round your answer
to the nearest whole number.
4. The lengths of the sides of a triangle are 5 cm,
8 cm, and 10 cm. Is it acute, right, or obtuse? 15
25 mi
obtuse
8-1