Question

1. what is x in the diagram below?

options: 2√10, 10√14, 20√3, 10√4

Explanation:

Step1: Recall the geometric mean theorem (altitude-on-hypotenuse theorem) for right triangles. The theorem states that in a right triangle, the altitude drawn to the hypotenuse is the geometric mean of the lengths of the two segments it divides the hypotenuse into. So, if the two segments are \( a = 4 \) and \( b = 10 \), and the altitude is \( x \), then \( x=\sqrt{a\times b} \).

Step2: Substitute the values of \( a \) and \( b \) into the formula. Here, \( a = 4 \) and \( b = 10 \), so \( x=\sqrt{4\times10}=\sqrt{40} \). Simplify \( \sqrt{40} \): \( \sqrt{4\times10}=\sqrt{4}\times\sqrt{10}=2\sqrt{10} \).

Answer:

\( 2\sqrt{10} \) (corresponding to the first option: \( 2\sqrt{10} \))