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.
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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: \[ x^2 = 7225 \implies x = \sqrt{7225} = 85 \] -
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.