Question

An isosceles triangle, triangle ABC, has the measurements AB = BC = 12. The segment BH is the altitude of triangle ABC, and BH = AH = CH = x. Which of the following correctly applies the Pythagorean Theorem to find x?
A. x = 6.
B. x = 72.
C. x = 12 square root 2.
D. x = 6 square root 2.

Answer 1

To solve for \( x \) in triangle \( ABC \) with \( AB = BC = 12 \) and \( BH \) being the altitude, we can start by determining the lengths of its other sides using the properties of the isosceles triangle.

Since \( AB = BC \), we can drop a perpendicular from point \( B \) to side \( AC \) at point \( H \), which also bisects \( AC \) because the triangle is isosceles.

Let's denote the following:
- \( AH = CH = x \) (the segments on side \( AC \))
- \( AC = AH + CH = x + x = 2x \)

In right triangle \( ABH \), we can apply the Pythagorean theorem:

\[
AB^2 = AH^2 + BH^2
\]

Substituting the known values, we have:

\[
12^2 = x^2 + x^2
\]

This simplifies to:

\[
144 = 2x^2
\]

Dividing both sides by 2 gives:

\[
72 = x^2
\]

Taking the square root of both sides results in:

\[
x = \sqrt{72} = 6\sqrt{2}
\]

So, the correct answer is:

**D. \( x = 6\sqrt{2} \)**