Question

solve for x (hint: draw 3 triangles)
answer attempt 2 out of 2

Explanation:

Step1: Recall geometric mean theorem

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. Also, each leg is the geometric mean of the hypotenuse and the adjacent segment. Let the altitude be \( h = 12 \), one segment be \( 16 \), and the other be \( x \). The leg adjacent to \( x \) and the altitude form similar triangles, so we use the geometric mean: \( x=\frac{12^{2}}{16} \) (Wait, no, actually, the geometric mean theorem states that \( h^{2}=x\times16 \)? Wait, no, let's correct. The geometric mean theorem (altitude-on-hypotenuse theorem) states that in a right triangle, the length of the altitude drawn from the right angle to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse. Also, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Wait, actually, the three triangles: the big right triangle, and the two smaller right triangles are all similar. So the triangle with leg \( x \) and altitude \( 12 \) is similar to the triangle with leg \( 12 \) and segment \( 16 \)? Wait, no. Let's denote the big triangle as \( \triangle ABC \), right-angled at \( A \), with altitude \( AD \) to hypotenuse \( BC \), where \( BD = x \), \( DC = 16 \), \( AD = 12 \). Then, \( \triangle ABD \sim \triangle CAD \sim \triangle ABC \). So from \( \triangle ABD \sim \triangle CAD \), we have \( \frac{BD}{AD}=\frac{AD}{DC} \), so \( \frac{x}{12}=\frac{12}{16} \). Then cross-multiplying: \( 16x = 12\times12 \), so \( x=\frac{144}{16}=9 \). Wait, that's the correct approach. So the key is the similarity of the right triangles. So the two smaller right triangles are similar, so the ratio of corresponding sides is equal. So \( \frac{x}{12}=\frac{12}{16} \).

Step2: Solve for \( x \)

From \( \frac{x}{12}=\frac{12}{16} \), cross - multiply: \( 16x = 12\times12 \)
Calculate \( 12\times12 = 144 \), so \( 16x = 144 \)
Then \( x=\frac{144}{16}=9 \)

Answer:

\( x = 9 \)