Question

Question content area top left
Part 1
Find the length of the third side of the right triangle.
.
.
.
Question content area top right
Part 1
    
5
13
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 given two sides, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).

Given:
- One side (leg) = 5
- The other side (hypotenuse) = 13
- The third side (let's say, b) is the other leg we need to find.

According to the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where \( a \) and \( b \) are the legs of the triangle and \( c \) is the hypotenuse.

Plugging in the known values:
\[
5^2 + b^2 = 13^2
\]

Calculating the squares:
\[
25 + b^2 = 169
\]

Now, isolate \( b^2 \):
\[
b^2 = 169 - 25
\]
\[
b^2 = 144
\]

Taking the square root of both sides gives:
\[
b = \sqrt{144}
\]
\[
b = 12
\]

Therefore, the length of the third side is:
\[
\boxed{12}
\]