Download 8-2 The Pythagorean Theorem and Its Converse

THE PYTHAGOREAN THEOREM AND ITS CONVERSE
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?
1.
2.
3.
4.
Check Skills You’ll Need
8-2
THE PYTHAGOREAN THEOREM AND ITS CONVERSE
GEOMETRY LESSON 8-1
Solutions
1. 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; 9 + 16 = 25
2. 52 = (5)(5) = 25; 122 = (12)(12) = 144; 132 = (13)(13) = 169;
25 + 144 = 169
3. 62 = (6)(6) = 36; 82 = (8)(8) = 64; 102 = (10)(10) = 100; 36 + 64 = 100
4. 42 = (4)(4) = 16; (4
2 )2 = (4 2)(4 2) = 16 (2)(2) = 16(2) = 32; 16 + 16 = 32.
8-2
THE PYTHAGOREAN THEOREM AND ITS CONVERSE
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
162 + 302 = c2
256 + 900 = c2
Use the Pythagorean Theorem.
Substitute 16 for a and 30 for b.
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.
8-2
Quick Check
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.
x2 + 102 = 122
Substitute x for a, 10 for b, and 12 for c.
x2 + 100 = 144
Simplify.
x2 = 44
x =
4(11)
x = 2 11
Subtract 100 from each side.
Take the square root of each side.
Simplify.
8-2
Quick Check
THE PYTHAGOREAN THEOREM AND ITS CONVERSE
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 = c2
902 + 902 = c2
8100 + 8100 = c2
Use the Pythagorean Theorem.
Substitute 90 for a and for b.
Simplify.
16,200 = c2
c=
c
16,200
Take the square root.
127.27922 Use a calculator.
The distance to home plate from second base is about 127 ft.
Quick Check
8-12
THE PYTHAGOREAN THEOREM AND ITS CONVERSE
GEOMETRY LESSON 8-1
Is this triangle a right triangle?
a2 + b2 c2
42 + 62
16 + 36
72
49
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
THE PYTHAGOREAN THEOREM AND ITS CONVERSE
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 + b2
Compare c2 with a2 + b2.
252
152 + 202
Substitute the greatest length for c.
625
225 + 400
Simplify.
625 = 625
Because c2 = a2 + b2, the triangle is a right triangle.
8-2
THE PYTHAGOREAN THEOREM AND ITS CONVERSE
GEOMETRY LESSON 8-1
(Continued)
b. 10, 15, 20
c2
a2 + b2
Compare c2 with a2 + b2.
202 102 + 152
400 100 + 225
400
Substitute the greatest length for c.
Simplify.
325
Because c2
a2 + b2, the triangle is obtuse.
Quick Check
8-2
THE PYTHAGOREAN THEOREM AND ITS CONVERSE
GEOMETRY LESSON 8-1
1. Find the value of x.
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.
25 mi
15
2. Find the value of x. Leave your
answer in simplest radical form.
4. The lengths of the sides of a triangle are 5 cm,
8 cm, and 10 cm. Is it acute, right, or obtuse?
obtuse
2
33
8-2