Question

5 (03.06 mc)
what expression represents the value of v? (2 points)
○ $v = \sqrt{zy}$
○ $v = \sqrt{wz}$
○ $v = \sqrt{w(w + z)}$
○ $v = \sqrt{z(w + z)}$

Explanation:

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

In a right - triangle \(ABC\) with right - angle at \(B\) and altitude \(BD\) to the hypotenuse \(AC\) (\(AC=w + z\)), for the right - triangle \(ABD\) (right - angled at \(D\)) and the large right - triangle \(ABC\) (right - angled at \(B\)).
The geometric mean theorem states that if in a right - triangle, a perpendicular is drawn from the right - angle vertex to the hypotenuse, then the length of a leg of the right - triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.
For leg \(AB = v\), hypotenuse \(AC=w + z\) and the segment of the hypotenuse adjacent to \(AB\) is \(z\).
The formula from the geometric mean theorem is \(v^{2}=z(w + z)\) (where \(v\) is the length of the leg, \(z\) is the adjacent segment of the hypotenuse, and \(w + z\) is the length of the hypotenuse).

Step2: Solve for \(v\)

Take the square root of both sides of the equation \(v^{2}=z(w + z)\).
Using the property \(\sqrt{a^{2}}=\vert a\vert\), and since \(v>0\) (as \(v\) represents the length of a side of a triangle), we get \(v=\sqrt{z(w + z)}\).

Answer:

\(v=\sqrt{z(w + z)}\) (the fourth option)