Question

What is the value of x
if 13, 84, x
is a Pythagorean triple?(1 point)
Responses

94
94

85
85

97
97

86

Answer 1

To find the value of \( x \) that makes \( 13, 84, x \) a Pythagorean triple, we can use the Pythagorean theorem, which states:

\[
a^2 + b^2 = c^2
\]

where \( c \) is the largest number and \( a \) and \( b \) are the other two sides of the triangle.

In this case, we need to identify which of the three numbers \( 13, 84, x \) is the largest. Since \( x \) is unknown, we will consider two scenarios: where \( x \) is the largest or one of the other two is the largest.

1. **Case 1: \( x \) is the largest.**
Then we have:
\[
13^2 + 84^2 = x^2
\]
Calculating:
\[
13^2 = 169
\]
\[
84^2 = 7056
\]
Adding these:
\[
169 + 7056 = 7225
\]
Therefore:
\[
x^2 = 7225 \implies x = \sqrt{7225} = 85
\]

2. **Case 2: \( 84 \) is the largest.**
Then:
\[
13^2 + x^2 = 84^2
\]
We already calculated \( 13^2 = 169 \) and \( 84^2 = 7056 \):
\[
169 + x^2 = 7056
\]
Rearranging gives:
\[
x^2 = 7056 - 169
\]
\[
x^2 = 6887
\]
Taking the square root:
\[
x = \sqrt{6887} \approx 82.9
\]
(not a valid integer).

Since the only valid integer value of \( x \) is from Case 1 (where \( x = 85 \)), we conclude that:

The value of \( x \) is **85**.