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(D) In $\Delta ABC$,$m \angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$.According to the geometric mean theorem for righ

Vedclass · Accepted Answer

$16$

Vedclass · Answer

(D) In $\Delta ABC$,$m \angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$.According to the geometric mean theorem for right-angled triangles,the altitude to the hypotenuse divides the triangle into two triangles similar to the original triangle and to each other.Specifically,$BM^2 = AM \cdot CM$.Given $AM = 8$ and $BM = 8$,we substitute these values into the equation:$8^2 = 8 \cdot CM$$64 = 8 \cdot CM$$CM = \frac{64}{8} = 8$.Now,the length of the hypotenuse $AC$ is the sum of the segments $AM$ and $CM$:$AC = AM + CM = 8 + 8 = 16$.Therefore,the correct option is $D$.

In $\Delta ABC$,$m \angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $AM = BM = 8$,then $AC = \ldots$

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