Question

1/5 9 16 x 10 12 14 16

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. Let the hypotenuse be divided into segments of length \( a = 9 \) and \( b = 16 \), and the altitude be \( x \). The formula is \( x=\sqrt{a\times b} \)? Wait, no, actually the geometric mean theorem (altitude-on-hypotenuse theorem) states that \( x^{2}=9\times16 \)? Wait, no, wait. Wait, the two segments of the hypotenuse are 9 and 16? Wait, no, looking at the triangle: the hypotenuse is split into two parts, one is 9, the other is 16? Wait, no, the triangle has a hypotenuse with segments 9 and 16? Wait, no, the right triangle, when we draw the altitude to the hypotenuse, the length of the altitude \( x \) satisfies \( x^{2}= \) (length of one segment) \( \times \) (length of the other segment). Wait, actually, the formula is: if the hypotenuse is divided into segments of length \( m \) and \( n \), then the altitude \( h \) (here \( x \)) is given by \( h=\sqrt{m\times n} \)? Wait, no, \( h^{2}=m\times n \). Wait, let's check: in a right triangle, the altitude to the hypotenuse creates two smaller similar triangles, each similar to the original triangle and to each other. So, the ratio of corresponding sides: \( \frac{x}{9}=\frac{16}{x} \), so cross - multiplying gives \( x^{2}=9\times16 \)? Wait, no, wait, maybe I mixed up the segments. Wait, the two segments of the hypotenuse: one is 9, the other is 16? Wait, the hypotenuse is 9 + 16? No, wait, the triangle: the side adjacent to the 16 segment and the 9 segment? Wait, no, the diagram: the hypotenuse is split into two parts, one is 9, the other is 16? Wait, no, the right triangle, with the altitude \( x \) to the hypotenuse. The two segments of the hypotenuse are 9 and 16? Wait, no, actually, the formula is \( x = \sqrt{9\times16} \)? Wait, no, \( x^{2}=9\times16 \), so \( x=\sqrt{144}=12 \). Wait, let's do it step by step.

Step1: Apply the geometric mean theorem

In a right triangle, when an altitude (\( x \)) is drawn to the hypotenuse, the length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse. Let the two segments of the hypotenuse be \( a = 9 \) and \( b = 16 \). Then, by the geometric mean theorem, \( x^{2}=a\times b \).

Step2: Substitute the values

Substitute \( a = 9 \) and \( b = 16 \) into the formula: \( x^{2}=9\times16 \). Calculate \( 9\times16 = 144 \).

Step3: Solve for \( x \)

Take the square root of both sides: \( x=\sqrt{144}=12 \).

Answer:

12