Question

Question content area top left
Part 1
Find the length of the third side of the right triangle.
.
.
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Question content area top right
Part 1
    
6
10
b
        
Question content area bottom
Part 1
The length of the third side is
  
enter your response here.
​(Simplify your answer. Type an exact​ answer, using radicals as​ needed.)

Answer 1

To find the length of the third side of a right triangle when the lengths of the other two sides are given, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Given:

- One side (leg) = 6
- The other side (leg) = 10
- The hypotenuse = \( b \)

According to the Pythagorean theorem:

\[
b^2 = 6^2 + 10^2
\]

Calculating the squares:

\[
b^2 = 36 + 100
\]
\[
b^2 = 136
\]

Now, to find \( b \):

\[
b = \sqrt{136}
\]

Next, we simplify \( \sqrt{136} \):

\[
\sqrt{136} = \sqrt{4 \times 34} = \sqrt{4} \times \sqrt{34} = 2\sqrt{34}
\]

So, the length of the hypotenuse (which is the third side) is:

\[
b = 2\sqrt{34}
\]

Thus, the length of the third side is

\[
\boxed{2\sqrt{34}}
\]