Question

Solve for X on a triangle with three sides, 1 side is 36 cm, the second side is 77 cm and the third side is x, calculate x using the Pythagorean Theorem

Answer 1

To solve for \( x \) using the Pythagorean Theorem, we first need to determine if we have a right triangle. The Pythagorean Theorem 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).

In this case, we have three sides:
- \( a = 36 \, \text{cm} \)
- \( b = 77 \, \text{cm} \)
- \( c = x \)

To apply the theorem, we need to assume two scenarios: either \( x \) is the longest side (hypotenuse), or one of the other two sides is the hypotenuse. We will investigate the first scenario where \( x \) is the hypotenuse, as it is likely to be the case.

### Scenario 1: \( x \) is the hypotenuse
According to the Pythagorean theorem:
\[
x^2 = 36^2 + 77^2
\]

Calculating the squares:
\[
x^2 = 1296 + 5929
\]
\[
x^2 = 7225
\]

Now, take the square root to find \( x \):
\[
x = \sqrt{7225} = 85 \, \text{cm}
\]

### Scenario 2: \( x \) is one of the legs
Next, we can check if \( x \) could be a leg of the triangle. Here, we would consider \( x \) as one of the legs and either \( 36 \, \text{cm} \) or \( 77 \, \text{cm} \) as the hypotenuse. But since \( x \) represents an unknown side, we can look at the cases when either leg is used as the hypotenuse.

However, given \( 77 > 36 \), we check:
1. If \( 77 \) is the hypotenuse:
\[
77^2 = 36^2 + x^2
\]
\[
5929 = 1296 + x^2
\]
\[
x^2 = 5929 - 1296 = 4633
\]
\[
x = \sqrt{4633} \approx 68.06 \, \text{cm}
\]

### Conclusion
- In the scenario where \( x \) is the hypotenuse, \( x = 85 \, \text{cm} \).
- In the scenario where \( 77 \, \text{cm} \) is the hypotenuse, \( x \approx 68.06 \, \text{cm} \).

The value of \( x \) depends on whether it is the hypotenuse or one of the legs. The two calculated lengths are \( 85 \, \text{cm} \) (if \( x \) is the hypotenuse) and approximately \( 68.06 \, \text{cm} \) (if \( 77 \, \text{cm} \) is the hypotenuse). If you need \( x \) to be a specific leg or side, additional context about which sides are which would be required.