Question

find the exact value of y. y =

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. So, if we have a right triangle with altitude \( h \) (here \( h = 18 \)) and segments \( a = 12 \) and \( b = y \), then \( h^2=a\times b \).

Step2: Substitute Values and Solve for \( y \)

Substitute \( h = 18 \) and \( a = 12 \) into the formula \( h^2=a\times y \). We get \( 18^2=12\times y \). Calculate \( 18^2 = 324 \), so the equation becomes \( 324 = 12y \). Then, solve for \( y \) by dividing both sides by 12: \( y=\frac{324}{12} \). Simplify the fraction: divide numerator and denominator by 12's greatest common divisor with 324, which is 12? Wait, 324 ÷ 12 = 27? Wait, no, 12×27 = 324? 12×20=240, 12×7=84, 240+84=324. Wait, but actually, the Geometric Mean Theorem (or Altitude-on-Hypotenuse Theorem) states that in a right triangle, the length of the altitude drawn to the hypotenuse is the geometric mean between 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, maybe I misapplied. Let's look at the triangle: there is a right triangle, with an altitude drawn to the hypotenuse, creating two smaller right triangles. So the triangle with legs 12 and 18, and the other triangle with leg \( y \) and 18. Wait, actually, the two smaller triangles are similar to the original triangle and to each other. So the ratio of corresponding sides should be equal. So the triangle with base 12 and height 18 (the altitude) is similar to the triangle with base \( y \) and height 18? Wait, no. Let's denote the original right triangle as \( \triangle ABC \), right - angled at \( C \), with altitude \( CD \) to hypotenuse \( AB \), where \( AD = 12 \), \( CD = 18 \), and \( DB = y \). Then, by the Geometric Mean Theorem, \( CD^2=AD\times DB \). Yes, that's the correct formula. So \( 18^2=12\times y \). Then \( y=\frac{18^2}{12}=\frac{324}{12}=\frac{27\times12}{12}=27 \)? Wait, no, 324 ÷ 12: 12×27 = 324? 12×20 = 240, 12×7 = 84, 240 + 84 = 324. Wait, but let's check: 18×18 = 324, 12×27 = 324. So \( y=\frac{324}{12}=27 \)? Wait, but let's verify. If \( CD^2 = AD\times DB \), then \( DB=\frac{CD^2}{AD} \). So \( DB=\frac{18^2}{12}=\frac{324}{12}=27 \).

Answer:

\( y = 27 \)