To find the value of $x$ in the isosceles triangle:

Step-by-Step Solution:

1. Triangle Details:

   • The triangle is isosceles.
   • The two equal sides are $\sqrt{20}$.
   • The base of the triangle is $8$.
   • The height $x$ divides the base into two equal parts, so each half is $\frac{8}{2} = 4$.

   Now we have a right triangle where:

   • The hypotenuse is $\sqrt{20}$,
   • One leg is $x$ (height),
   • The other leg is $4$ (half the base).

2. Using the Pythagorean Theorem: The Pythagorean theorem states: $(\text{hypotenuse})^2 = (\text{leg}_1)^2 + (\text{leg}_2)^2$ Substituting values: $(\sqrt{20})^2 = x^2 + 4^2$ Simplify: $20 = x^2 + 16$

3. Solve for $x^2$: Subtract $16$ from both sides: $x^2 = 20 - 16$ $x^2 = 4$

4. Find $x$: Take the square root of both sides: $x = \sqrt{4} = 2$

Final Answer:

The value of $x$ is 2.

The correct option is: C) $x = 2$.

Do you need a detailed explanation for any step?

Here are 5 related questions to expand your understanding:

1. How does the Pythagorean theorem work in right triangles?
2. What are the properties of isosceles triangles?
3. How do you calculate the height of a triangle with given sides and base?
4. If the hypotenuse and one leg are given, how do you find the other leg?
5. What are the special right triangles, such as $45^\circ$-$45^\circ$-$90^\circ$ triangles?

Tip: In isosceles triangles, the height always bisects the base and divides the triangle into two congruent right triangles.