Question

question 8 (1 point)
find the value of x.
triangle with segments x, 9, and altitude 12, right angles at the altitudes intersections

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. So, \( x \) and \( 9 \) are segments of the hypotenuse, and the altitude is \( 12 \). By the geometric mean theorem, \( x \times (x + 9)=(x + 9)\) related? Wait, no, the correct relation is that if a leg (altitude) is \( 12 \), and the segments are \( x \) and \( 9 \), then \( 12^2=x\times9 \)? Wait, no, wait. Wait, the altitude to the hypotenuse: in a right triangle, the length of the altitude \( h \) to the hypotenuse is the geometric mean of the lengths of the two segments \( p \) and \( q \) that the hypotenuse is divided into. So \( h^2 = p\times q \). But also, each leg is the geometric mean of the hypotenuse and the adjacent segment. Wait, the triangle here: the big triangle is right-angled, and there's an altitude to the hypotenuse, dividing the hypotenuse into \( x \) and \( 9 \), and the altitude is \( 12 \). Wait, no, actually, the two smaller triangles are similar to the big triangle and to each other. So, the leg of the big triangle (the one adjacent to segment \( x \)) is the geometric mean of \( x \) and the entire hypotenuse? Wait, no, let's label the triangle. Let the big right triangle have hypotenuse \( c=x + 9 \), and the altitude to the hypotenuse is \( 12 \). Then, by the geometric mean theorem (altitude-on-hypotenuse theorem), \( 12^2=x\times9 \)? Wait, no, that's not right. Wait, no, the altitude is \( 12 \), and the two segments of the hypotenuse are \( x \) and \( 9 \). Then, the altitude squared is equal to the product of the two segments. Wait, yes! The altitude to the hypotenuse: \( h^2=p\times q \), where \( p \) and \( q \) are the segments. Wait, no, wait, no. Wait, the legs: each leg is the geometric mean of the hypotenuse and the adjacent segment. Wait, maybe I mixed up. Let's correct: In a right triangle, if you draw an altitude from the right angle to the hypotenuse, then:

1. The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse. So \( h=\sqrt{p\times q} \), so \( h^2 = p\times q \).

2. Each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So, if one leg is \( a \), adjacent segment is \( p \), hypotenuse is \( c \), then \( a^2=c\times p \).

But in this problem, the altitude is \( 12 \), and the two segments of the hypotenuse are \( x \) and \( 9 \). Wait, no, looking at the diagram: the big triangle is right-angled, and there's an altitude (the red line) to the hypotenuse, creating two smaller right triangles. The segment of the hypotenuse adjacent to \( x \) is \( x \), and the other segment is \( 9 \), and the altitude is \( 12 \). Wait, no, actually, the side labeled \( x \) is a segment of the hypotenuse, and the other segment is \( 9 \), and the altitude is \( 12 \). Then, by the altitude-on-hypotenuse theorem, \( 12^2=x\times9 \)? Wait, no, that would be if the altitude is between \( x \) and \( 9 \). Wait, no, maybe the leg is \( 12 \)? Wait, no, the red line is the altitude, length \( 12 \), and the two segments of the hypotenuse are \( x \) and \( 9 \). Then, the altitude squared is equal to the product of the two segments. So \( 12^2=x\times9 \)? Wait, that would give \( x = 144/9 = 16 \). Wait, let's check.

Wait, the altitude-on-hypotenuse theorem states that in a right triangle, the altitude drawn to the hypotenuse is the geometric mean between the segments of the hypotenuse. So, if the hypotenuse is divided into segments of length \( p \) and \( q \), and the altitude is \( h \), then \( h^2 = p \times q \). So here, \( h = 12 \), \( p = x \), \( q = 9 \). So \( 12^2 = x \times 9 \).

Step2: Solve for \( x \)

Calculate \( 12^2 = 144 \), so \( 144 = 9x \). Then, divide both sides by \( 9 \): \( x=\frac{144}{9}=16 \).

Answer:

\( 16 \)