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?

Question

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