Question

applying the geometric mean (leg) theorem
what is the value of q?
4√5
20√5
64√5
2√14

Explanation:

Step1: Recall Geometric Mean (Leg) Theorem

In a right triangle, the length of a leg is the geometric mean of the length of the hypotenuse segment adjacent to that leg and the length of the entire hypotenuse. For right triangle \( \triangle QSR \) with right angle at \( S \) and altitude \( ST \) to hypotenuse \( QR \), the theorem states \( q^2 = RT \times QR \).
First, find the length of \( QR \): \( QR = QT + RT = 10 + 4 = 14 \)? Wait, no, wait. Wait, \( QT = 10 \), \( RT = 4 \), so \( QR = 10 + 4 = 14 \)? Wait, no, wait the Geometric Mean (Leg) Theorem: the leg \( q \) (which is \( SR \)) is the geometric mean of \( RT \) and \( QR \)? Wait, no, let's correct. The Geometric Mean (Leg) Theorem: In a right triangle, when an altitude is drawn to the hypotenuse, each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So for leg \( SR = q \), the adjacent segment is \( RT = 4 \), and the hypotenuse \( QR = QT + RT = 10 + 4 = 14 \)? Wait, no, wait the diagram: \( QT = 10 \), \( RT = 4 \), so \( QR = 10 + 4 = 14 \). Then by the theorem, \( q^2 = RT \times QR = 4 \times 14 \)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, the two segments of the hypotenuse are \( QT = 10 \) and \( RT = 4 \), so the hypotenuse \( QR = 10 + 4 = 14 \). Then the leg \( q \) (SR) is the geometric mean of \( RT \) and \( QR \)? Wait, no, the correct formula is: if the hypotenuse is divided into segments of length \( a \) and \( b \), and the legs are \( l_1 \) and \( l_2 \), then \( l_1^2 = a \times (a + b) \) and \( l_2^2 = b \times (a + b) \)? Wait, no, no. Wait, the Geometric Mean (Leg) Theorem: In a right triangle, the length of a leg is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg. So for leg \( SR \) (which is \( q \)), the adjacent segment is \( RT = 4 \), and the hypotenuse is \( QR = QT + RT = 10 + 4 = 14 \). Wait, but that would be \( q^2 = RT \times QR = 4 \times 14 = 56 \), so \( q = \sqrt{56} = 2\sqrt{14} \)? No, that's not matching. Wait, maybe I got the segments wrong. Wait, maybe \( QT = 10 \), \( RT = 4 \), so the hypotenuse is \( QR = 10 + 4 = 14 \), but the other leg: wait, no, maybe the altitude is \( ST \), and the two triangles \( \triangle STR \) and \( \triangle QSR \) are similar. So \( \triangle STR \sim \triangle QSR \), so \( \frac{SR}{QR} = \frac{RT}{SR} \), so \( SR^2 = RT \times QR \). Wait, \( RT = 4 \), \( QR = 10 + 4 = 14 \), so \( SR^2 = 4 \times 14 = 56 \), \( SR = \sqrt{56} = 2\sqrt{14} \)? But that's one of the options. Wait, but wait, maybe I made a mistake. Wait, no, let's check again. Wait, the Geometric Mean (Leg) Theorem: Each leg of a right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So leg \( q \) (SR) is adjacent to segment \( RT = 4 \), hypotenuse \( QR = 10 + 4 = 14 \). So \( q^2 = 4 \times 14 = 56 \), so \( q = \sqrt{56} = 2\sqrt{14} \)? Wait, but the first option is \( 4\sqrt{5} \), which is \( \sqrt{16 \times 5} = \sqrt{80} \), no. Wait, maybe I mixed up the segments. Wait, maybe \( QT = 10 \), \( RT = 4 \), so the hypotenuse is \( QR = 10 + 4 = 14 \), but the other leg: wait, no, maybe the altitude is \( ST \), and the two segments are \( QT = 10 \) and \( RT = 4 \), so the leg \( q \) (SR) is the geometric mean of \( RT \) and \( QR \)? Wait, no, the correct formula is \( leg^2 = segment \times hypotenuse \). So if \( q \) is the leg, adjacent segment is \( RT = 4 \), hypotenuse \( QR = 10 + 4 = 14 \), so \( q^2 = 4 \times 14 = 56 \), \( q = \sqrt{56} = 2\sqrt{14} \), which is the last option. Wait, but let's check the options again. The options are \( 4\sqrt{5} \), \( 20\sqrt{5} \), \( 64\sqrt{5} \), \( 2\sqrt{14} \). So \( 2\sqrt{14} \) is an option. Wait, but maybe I made a mistake in the segments. Wait, maybe \( QT = 10 \), \( RT = 4 \), so the hypotenuse is \( QR = 10 + 4 = 14 \), but the leg \( q \) is \( SR \), and the similar triangles: \( \triangle STR \sim \triangle QSR \), so \( \frac{SR}{QR} = \frac{RT}{SR} \), so \( SR^2 = RT \times QR = 4 \times 14 = 56 \), so \( SR = \sqrt{56} = 2\sqrt{14} \), which is the fourth option. Wait, but let's check the first option: \( 4\sqrt{5} \), \( (4\sqrt{5})^2 = 16 \times 5 = 80 \). If \( q^2 = 80 \), then \( RT \times QR = 80 \), so \( QR = 80 / 4 = 20 \), but \( QT + RT = 20 \), so \( QT = 16 \), but in the diagram \( QT = 10 \). So that's not matching. So the correct one should be \( 2\sqrt{14} \). Wait, but let's re-express the Geometric Mean (Leg) Theorem correctly. The theorem states that in a right triangle, when an altitude is drawn to the hypotenuse, each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So, if the hypotenuse is divided into segments of length \( x \) (adjacent to leg 1) and \( y \) (adjacent to leg 2), and the hypotenuse length is \( x + y \), then leg 1 (adjacent to \( x \)) satisfies \( leg1^2 = x \times (x + y) \), and leg 2 (adjacent to \( y \)) satisfies \( leg2^2 = y \times (x + y) \). In this problem, leg \( q \) (SR) is adjacent to segment \( RT = 4 \) (so \( y = 4 \)), and the hypotenuse \( QR = QT + RT = 10 + 4 = 14 \) (so \( x + y = 14 \), where \( x = QT = 10 \), \( y = RT = 4 \)). Wait, no, actually, the leg adjacent to \( RT \) is \( SR = q \), so \( q^2 = RT \times QR = 4 \times 14 = 56 \), so \( q = \sqrt{56} = 2\sqrt{14} \), which is the fourth option. So that's the correct answer.

Step2: Calculate \( q \)

Using the Geometric Mean (Leg) Theorem: \( q^2 = RT \times QR \).
\( RT = 4 \), \( QR = QT + RT = 10 + 4 = 14 \).
So \( q^2 = 4 \times 14 = 56 \).
Take the square root: \( q = \sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14} \).

Answer:

\( 2\sqrt{14} \) (the fourth option, or the one labeled \( 2\sqrt{14} \))