Question

right triangle with lengths of 6 and 12 what is the missing length

Answer 1

In a right triangle, you can use the Pythagorean theorem to find the missing length. The theorem states that for a right triangle with legs of lengths \(a\) and \(b\), and hypotenuse \(c\):

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

Assuming the lengths 6 and 12 are the lengths of the legs of the triangle, you can calculate the hypotenuse as follows:

1. Let \(a = 6\) and \(b = 12\).
2. Substitute the values into the equation:

\[
c^2 = 6^2 + 12^2
\]
\[
c^2 = 36 + 144
\]
\[
c^2 = 180
\]

3. Now take the square root to find \(c\):

\[
c = \sqrt{180}
\]
\[
c = \sqrt{36 \cdot 5} = 6\sqrt{5} \approx 13.42
\]

So if both 6 and 12 are legs of the triangle, the hypotenuse is approximately \(13.42\).

If one of the lengths provided (6 or 12) is the hypotenuse, and you need to find the length of the other leg, you can adjust accordingly:

### Case 1: If \(c = 12\) (hypotenuse) and \(a = 6\) (one leg):

\[
12^2 = 6^2 + b^2
\]
\[
144 = 36 + b^2
\]
\[
b^2 = 144 - 36
\]
\[
b^2 = 108
\]
\[
b = \sqrt{108} = 6\sqrt{3} \approx 10.39
\]

### Conclusion:
- If 6 and 12 are the legs, the hypotenuse is \(6\sqrt{5} \approx 13.42\).
- If 12 is the hypotenuse and 6 is one leg, the other leg is \(6\sqrt{3} \approx 10.39\).