9.4 The Pythagorean Theorem 749EXAMPLE (^3) The lengths of two sides of a right triangle
are given in the figure. Find the missing side length.
StrategyWe will use the Pythagorean theorem to find the
missing side length.
WHYIf we know the lengths of any two sides of a right triangle,
we can find the length of the third side using the Pythagorean
theorem.
SolutionWe may substitute 11 for either or , but 61 must be
substituted for the length of the hypotenuse. If we choose to
substitute 11 for , we can find the unknown side length as
follows.
This is the Pythagorean equation.
Substitute 11 for band 61 for c.
Evaluate each exponential expression.
To isolate a^2 on the left side, subtract
121 from both sides.
Do the subtraction.
If a^2 3,600, then amust be a square root of 3,600.
Because arepresents a length, it must be the positive
square root of 3,600.
Use a calculator, if necessary, to find the square root.
The missing side length is 60 ft.
a 60
a 1 3,600
a^2 3,600
a^2 121 121 3,721 121
a^2 121 3,721
a^2 112 612
a^2 b^2 c^2
b a
c
a b
Self Check 3
The lengths of two sides of a
right triangle are given. Find the
missing side length.
11 ft
61 ft
b = 11 ft
c = 61 ft a
Now TryProblem 23
65 in.
33 in.
2 Use the Pythagorean theorem to approximate
the length of a side of a right triangle.
When we use the Pythagorean theorem to find the length of a side of a right triangle,
the solution is sometimes the square root of a number that is not a perfect square. In
that case, we can use a calculator to approximatethe square root.
EXAMPLE (^4) Refer to the right triangle shown
here. Find the missing side length. Give the exact answer
and an approximation to the nearest hundredth.
StrategyWe will use the Pythagorean theorem to find the missing side length.
WHYIf we know the lengths of any two sides of a right triangle, we can find the
length of the third side using the Pythagorean theorem.
SolutionWe may substitute 2 for either or , but 6
must be substituted for the length of the hypotenuse.
If we choose to substitute 2 for , we can find the
unknown side length as follows.
This is the Pythagorean equation.
Substitute 2 for aand 6 for c.
Evaluate each exponential expression.
To isolate b^2 on the left side, undo the addition of 4 by
subtracting 4 from both sides.
b^2 32 Do the subtraction.
4 b^2 4 36 4
4 b^2 36
22 b^2 62
a^2 b^2 c^2
b
a
c
a b
Self Check 4
Refer to the triangle below.
Find the missing side length.
Give the exact answer and an
approximation to the nearest
hundredth.
Now TryProblem 35
5 m
7 m
2 in.
6 in.
a = 2 in.
b
c = 6 in.
3,721
121
3,600