Question

What is the length of the hypotenuse of the triangle when xequals12​?
3 x5 x plus 6



A right triangle has a horizontal leg labeled 3 x and a vertical leg labeled 5 x plus 6.
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Part 1
The length of the hypotenuse is about
  
enter your response here.
​(Round to the nearest tenth as​ needed.)

Answer 1

To find the length of the hypotenuse of a right triangle with legs labeled \(3x\) and \(5x + 6\), we can use the Pythagorean theorem, which is:

\[
c = \sqrt{a^2 + b^2}
\]

where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the legs.

1. Substitute \(x = 12\) into the expressions for the legs:
- For the horizontal leg:
\[
a = 3x = 3(12) = 36
\]
- For the vertical leg:
\[
b = 5x + 6 = 5(12) + 6 = 60 + 6 = 66
\]

2. Now, substitute \(a\) and \(b\) into the Pythagorean theorem:
\[
c = \sqrt{(36)^2 + (66)^2}
\]

3. Calculate \(36^2\) and \(66^2\):
\[
36^2 = 1296
\]
\[
66^2 = 4356
\]

4. Add them together:
\[
1296 + 4356 = 5652
\]

5. Finally, take the square root to find the length of the hypotenuse:
\[
c = \sqrt{5652} \approx 75.3
\]

Thus, the length of the hypotenuse is approximately **75.3** when rounded to the nearest tenth.