Question

in the diagram, the length of (overline{yz}) is twice the length of (overline{az}). (overline{ya}) is an altitude of (\triangle xyz). what is the length of (overline{ya})?
5(sqrt{3}) units
10(sqrt{3}) units
15 units
20 units

Explanation:

Step1: Find the length of \( AZ \)

Given \( YZ = 2AZ \) and \( YZ=10 \), then \( 10 = 2AZ \), so \( AZ = 5 \).

Step2: Use the geometric mean theorem (altitude - on - hypotenuse theorem)

In a right - triangle \( \triangle XYZ \) with altitude \( YA \) to the hypotenuse \( XZ \), we know that \( YA^{2}=AZ\times ZY \).
Substitute \( AZ = 5 \) and \( ZY = 10 \) into the formula: \( YA^{2}=5\times10 \). Wait, no, actually, in right - triangle \( \triangle YAZ \) (since \( \angle YAZ = 90^{\circ}\) and \( \angle YZY = 60^{\circ}\), \( \angle AYZ=30^{\circ}\)).
Another way: In right - triangle \( \triangle YAZ \), if \( \angle YZY = 60^{\circ}\) (because \( YZ = 2AZ \), in a \(30 - 60-90\) triangle, the side opposite the \(30^{\circ}\) angle is half the hypotenuse). Let \( AZ = x \), \( YZ = 2x\), and using the Pythagorean theorem \(YA=\sqrt{YZ^{2}-AZ^{2}}\).
Since \( YZ = 10\), \( AZ = 5\), then \(YA=\sqrt{10^{2}-5^{2}}=\sqrt{(10 + 5)(10 - 5)}=\sqrt{15\times5}=\sqrt{75}=5\sqrt{3}\).

Answer:

\(5\sqrt{3}\) units