Question

what is the value of q?
4√5
2√14
20√5
64√5

Explanation:

Step1: Apply the geometric mean theorem (altitude-on-hypotenuse theorem)

For a right - angled triangle formed by an altitude to the hypotenuse, the following relationship holds: If we have a triangle with segments of the hypotenuse \(a = 10\) and \(b=4\), and the length of the side adjacent to \(b\) is \(q\), then \(q=\sqrt{(a + b)\times b}\) (derived from \(q^{2}=b\times(a + b)\) by the geometric mean in right - angled triangles).

Step2: Substitute the values of \(a\) and \(b\)

Substitute \(a = 10\) and \(b = 4\) into the formula \(q=\sqrt{(10 + 4)\times4}\). First, calculate \((10 + 4)\times4=14\times4 = 56\). Then \(q=\sqrt{56}\).

Step3: Simplify the square root

We can rewrite \(\sqrt{56}\) as \(\sqrt{4\times14}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 4\), \(b = 14\)), we get \(q = 2\sqrt{14}\).

Answer:

\(2\sqrt{14}\)