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=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 $\mathrm{△}ABC$, we note that $AB=BC=12$ and $BH$ is the altitude from $B$ to $AC$, with $AH=CH=x$.

Since $\mathrm{△}ABC$ is isosceles, $AC=AH+CH=x+x=2x$.

Now applying the Pythagorean Theorem in triangle $\mathrm{△}ABH$, we have:

$$A{B}^2=A{H}^2+B{H}^2$$

Plugging in the values:

$${12}^2=x^2+B{H}^2$$

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).