Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).

• A. $x=10,y=5\sqrt5$
• B. $x=3,y=4$
• C. $x=5\sqrt3,y=10$
• D. $x=5,y=5\sqrt2$

Answer: (C)

Verified step by step guidance

1. Identify the type of triangle: The given triangle is a right triangle with one angle measuring 30 degrees.
2. Recall the properties of a 30-60-90 triangle: In a 30-60-90 triangle, the side opposite the 30-degree angle is half the hypotenuse, and the side opposite the 60-degree angle is the hypotenuse times the square root of 3 divided by 2.
3. Use the property of the 30-60-90 triangle to find the hypotenuse (y): Since the side opposite the 30-degree angle is 5, the hypotenuse y is twice this length, so y = 2 * 5.
4. Calculate the length of the side opposite the 60-degree angle (x): Use the formula for the 30-60-90 triangle, where the side opposite the 60-degree angle is the hypotenuse times the square root of 3 divided by 2. Therefore, x = y * sqrt(3) / 2.
5. Simplify the expressions: Ensure that the expressions for x and y are fully simplified to find the missing sides of the triangle.