Question

solve for the variable 6. triangle with segments labeled 12, 8, and x, and a right angle

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. Here, we use the property that \( x/12 = 12/8 \) (from the geometric mean theorem for the leg and the segments of the hypotenuse).

Step2: Solve the proportion

Cross - multiply the proportion \( \frac{x}{12}=\frac{12}{8} \). So, \( 8x = 12\times12 \).
Calculate \( 12\times12 = 144 \), then the equation becomes \( 8x=144 \).
Divide both sides by 8: \( x=\frac{144}{8}=18 \).

Answer:

\( x = 18 \)