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?

Question

(1 point)
Responses

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

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

x=72
x equals 72

x=6
x equals 6

Answer 1

To solve for \( x \) in the isosceles triangle \( \triangle ABC \) where \( AB = BC = 12 \) and \( BH \) is the altitude, we can apply the Pythagorean theorem.

In an isosceles triangle, the altitude from the vertex angle (in this case, \( B \)) to the base (in this case, \( AC \)) will bisect the base into two equal segments. Let's denote the midpoint of \( AC \) as point \( D \).

Since \( AB = BC = 12 \) and \( AD = DC \), we can say:

\[ AD = DC = \frac{AC}{2} \]

Let \( h = BH = x \).

Using the Pythagorean theorem in \( \triangle ABD \), we have:

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

Substituting the known values, we get:

\[ 12^2 = \left(\frac{AC}{2}\right)^2 + x^2 \]

Since \( AC \) is not given directly, it can be found by analyzing \( AD = DC \).

Now, we want the relationship when \( AD = 6√2 \) (which will lead us to explore \( AC\)).

Hence, the correct choice using the Pythagorean theorem is:

\[ x = 12^2 - (6√2)^2 \]

This translates to:

\[ x = 144 - 72 \]

Finally,

\[ x = 72 \]

Thus, the correct response is:

x=72, x equals 72.