1. 8-2 The Pythagorean Theorem and Its Converse

2. The Pythagorean Theorem and Its Converse
May 9, 2003
GEOMETRY LESSON 8-1
(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
8-2

3. The Pythagorean Theorem and Its Converse
May 9, 2003
GEOMETRY LESSON 8-1
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; ( )2 = (4 2)(4 2) = (2)(2) = 16(2) = 32; = 32.
8-2

4. The Pythagorean Theorem and Its Converse
May 9, 2003
GEOMETRY LESSON 8-1
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
8-2

5. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
GEOMETRY LESSON 8-1
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
8-2

6. The Pythagorean Theorem and Its Converse
May 9, 2003
GEOMETRY LESSON 8-1
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.
= 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
8-12

7. The Pythagorean Theorem and Its Converse
May 9, 2003
GEOMETRY LESSON 8-1
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
8-2

8. The Pythagorean Theorem and Its Converse
May 9, 2003
GEOMETRY LESSON 8-1
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.
8-2

9. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
GEOMETRY LESSON 8-1
(Continued)
b. 10, 15, 20
c2 a2 + b Compare c2 with a2 + b2.
Substitute the greatest length for c.
Simplify.
Because c a2 + b2, the triangle is obtuse.
Quick Check
8-2

10. The Pythagorean Theorem and Its Converse
May 9, 2003
The Pythagorean Theorem and Its Converse
GEOMETRY LESSON 8-1
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
8-2