Question

An angle measures 45 degrees, the opposite side is x. The adjacent side is y and the hypotenuse is 5 radical 2. It is a right triangle. Find the side length of x. Use trigonometric ratios.

X= radical 2
X=5
X= 2.5
X=3

Answer 1

To find the length of the side \( x \) (the side opposite the 45-degree angle) in a right triangle where the hypotenuse is \( 5\sqrt{2} \), we can use the properties of a 45-45-90 triangle.

In a 45-45-90 triangle, the ratios of the lengths of the sides are as follows:
- Each of the legs (the opposite and adjacent sides) are equal.
- The length of the hypotenuse is \( \sqrt{2} \) times the length of each leg.

Let's denote:
- The length of each leg (the opposite side and the adjacent side) as \( x \).
- The length of the hypotenuse as \( h \).

From the properties of a 45-45-90 triangle:
\[
h = x\sqrt{2}
\]

Given that the hypotenuse \( h = 5\sqrt{2} \), we can set up the equation:
\[
5\sqrt{2} = x\sqrt{2}
\]

To solve for \( x \), divide both sides by \( \sqrt{2} \):
\[
5 = x
\]

So, the length of side \( x \) (the opposite side) is \( 5 \).

Therefore, the answer is:
\[
\boxed{5}
\]