Question

An isosceles triangle, △ABC, has the measurements AB=BC=12. The segment BH is the altitude of △ABC, and BH=AH=CH=x. Which of the following correctly applies the Pythagorean Theorem to find x?

(1 point)
Responses

x=72
x equals 72

x=62–√
x equals 6 Start Root 2 End Root

x=122–√
x equals 12 Start Root 2 End Root

x=6

Answer 1

To find the correct application of the Pythagorean Theorem for triangle \( \triangle ABC \), we note that \( AB = BC = 12 \) and \( BH \) is the altitude from \( B \) to \( AC \), with \( AH = CH = x \).

Since \( \triangle ABC \) is isosceles, \( AC = AH + CH = x + x = 2x \).

Now applying the Pythagorean Theorem in triangle \( \triangle ABH \), we have:

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

Plugging in the values:

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

Since \( BH = x \):

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

This becomes:

\[
144 = 2x^2
\]

Dividing by 2 gives:

\[
72 = x^2
\]

Thus:

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

Therefore, the correct response is:

**x = 6√2 (x equals 6 Start Root 2 End Root)**.