Exercise
Solve x2 = 4.
x=±2
Exercise
Solve x2 = –4.
no real solution
Exercise
Solve √ x = 4.
x = 16
Exercise
Solve √ x = –4.
no real solution
Exercise
Solve √ – x = 4.
x = –16
leg
leg
25 square
units
4
16 square
units
5 3
9 square
units
25 square
units
+
16 square
units
4
5 3
=
9 square
units
2
4
16
+
2
39
=
2
525
The Pythagorean Theorem
If the hypotenuse of a right
triangle has length c, and the
legs have lengths a and b,
then a2 + b2 = c2.
Example 1
Find the hypotenuse of a
right triangle with legs of 8
and 15.
c2 = a2 + b2
c2 = 289
c2 = 82 + 152 √ c2 = √ 289
c2 = 64 + 225
c = 17
Example 2
Find the hypotenuse of a
right triangle with legs of 6
and 7.
c2 = a2 + b2
c2 = 85
c2 = 62 + 72 √ c2 = √ 85
c2 = 36 + 49
c = √ 85 ≈ 9.2
Example
Find the hypotenuse of a
right triangle with legs of 9
and 12.
15
Example
Find the hypotenuse of a 1
right triangle with legs of 2
and √ 3 .
2
1
Example
Find the hypotenuse of a
right triangle with legs of 1
and 1.
√2
Example 3
Find the leg of a right triangle
whose hypotenuse is 16 and
other leg is 7.
a2 + 72 = 162 a2 + 49 = 256
a2 + 49 – 49 = 256 – 49
a2 = 207
a = √ 207 ≈ 14.4
Example
Find the length of a leg of a
right triangle whose
hypotenuse is 39 and whose
other leg is 15.
36
Example
Find the length of a leg of a
right triangle whose
hypotenuse is 20 and whose
other leg is 10.
√ 300 ≈ 17.3
Converse
The converse is the
statement resulting when the
“if” part and the “then” part
of a conditional statement are
switched.
Converse of the
Pythagorean Theorem
If a triangle has sides
a, b, and c, such that
a2 + b2 = c2, then the triangle
is a right triangle.
Example 4
Determine whether a triangle
with sides of 12, 35, and 37 is
a right triangle.
a2 + b2 = c2
yes
2
2
2
12 + 35 = 37
144 + 1,225 = 1,369
1,369 = 1,369
Example 5
Determine whether a triangle
with sides of 8, 12, and 14 is
a right triangle.
a2 + b2 = c2
no
2
2
2
8 + 12 = 14
64 + 144 = 196
208 ≠ 196
Example
Determine whether a triangle
with sides of 15, 18, and 22
is a right triangle.
no; 152 + 182 ≠ 222
Example
Determine whether a triangle
with sides of 16, 30, and 34
is a right triangle.
yes; 162 + 302 = 1,156 = 342
Exercise
A 16 ft. ladder leans up
against the side of a
building. If the base of the
ladder is 4 ft. from the base
of the building, how high up
the side of the building will
the ladder reach?
15.5 ft.
Exercise
A 200 ft. tower is braced to
the ground by a cable, from
a point 150 ft. above the
ground to a point 87 ft. from
the base of the tower. How
long is the cable?
173.4 ft.
Exercise
The distance between bases
on a baseball diamond is
90 ft. How far is it from home
plate to second base?
127.3 ft.
Exercise
An opening for a window is
23” wide, 54” tall, and 60”
diagonally. Is the opening
“square”; that is, do the
height and width form a right
angle?
no