To find the value of $x$ in this isosceles triangle, let's analyze the given information.

Step-by-Step Solution:

1. Identify the Triangle Parts:

   • This is an isosceles triangle, where two sides are equal: $x$ and $x$.
   • The base of the triangle is given as 10.
   • The height from the top vertex to the midpoint of the base is given as 12.

2. Split the Triangle:

   • Since the triangle is isosceles and the height bisects the base, the base is split into two equal segments of $5$ each (half of 10).
   • Now, we have two right triangles with:
     • One leg = 5 (half of the base)
     • The other leg = 12 (the height)
     • Hypotenuse = $x$ (the equal sides)

3. Apply the Pythagorean Theorem:

   • For one of the right triangles: $x^2 = 5^2 + 12^2$
   • Calculate each term: $x^2 = 25 + 144$ $x^2 = 169$
   • Take the square root of both sides: $x = \sqrt{169} = 13$

Answer:

The value of $x$ is $13$.

Would you like further details on the Pythagorean Theorem or other related concepts?

Here are five additional related questions:

1. How do you calculate the height of an isosceles triangle if only the side lengths are known?
2. How can you find the area of this isosceles triangle?
3. What properties make an isosceles triangle unique compared to other triangles?
4. How would you find the base length if only the side length $x$ and height were given?
5. Can the Pythagorean Theorem be applied to any triangle, or only to specific types?

Tip: In isosceles triangles, if a perpendicular line is drawn from the vertex angle to the base, it always bisects the base and creates two congruent right triangles.