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

The Geometric Mean (Leg) Theorem states that 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 \( QR \), we have \( q^2 = RT \times QR \).
First, find the length of \( QR \). We know \( QT = 10 \) and \( RT = 4 \), so \( QR=QT + RT=10 + 4 = 14 \)? Wait, no, wait. Wait, maybe I misread. Wait, looking at the triangle, \( QT = 10 \), \( RT = 4 \), so \( QR=10 + 4=14 \)? Wait, no, maybe the segments are \( QT = 10 \) and \( RT = 4 \), so the hypotenuse \( QR=10 + 4 = 14 \)? Wait, no, wait the Geometric Mean (Leg) Theorem: for leg \( SR = q \), the adjacent segment is \( RT = 4 \), and the hypotenuse is \( QR=QT + RT=10 + 4 = 14 \)? Wait, no, maybe I made a mistake. Wait, actually, the Geometric Mean (Leg) Theorem is: if in a right triangle, an altitude is drawn to the hypotenuse, then each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So, \( q^2=RT\times QR \), where \( QR = QT + RT=10 + 4 = 14 \)? Wait, no, that can't be. Wait, maybe \( QT = 10 \) and \( RT = 4 \), so \( QR = 10 + 4 = 14 \)? Wait, but then \( q^2=4\times14 = 56 \), \( q=\sqrt{56}=2\sqrt{14} \)? No, that's one of the options. Wait, no, wait maybe I mixed up the segments. Wait, let's re - examine the triangle. The right triangle is \( \triangle QSR \) with right angle at \( S \), and \( ST \) is the altitude to hypotenuse \( QR \). So, the two segments of the hypotenuse are \( QT = 10 \) and \( RT = 4 \). Then, for leg \( SR = q \), by the Geometric Mean (Leg) Theorem, \( q^2=RT\times QR \), where \( QR=QT + RT = 10 + 4=14 \)? Wait, no, that's incorrect. Wait, the correct formula is that each leg is the geometric mean of the hypotenuse and the adjacent segment. So, \( q^2=RT\times QR \), but \( QR = QT+RT = 10 + 4 = 14 \)? Wait, no, maybe \( QT = 10 \) and \( RT = 4 \), so \( QR=14 \), then \( q^2=4\times14 = 56 \), \( q = \sqrt{56}=2\sqrt{14} \)? But wait, another way: maybe the segments are \( QT = 10 \) and \( RT = 4 \), and the leg \( q \) is adjacent to \( RT = 4 \), and the hypotenuse is \( QR=10 + 4 = 14 \). Wait, but let's check the options. One of the options is \( 2\sqrt{14} \), another is \( 4\sqrt{5} \), \( 20\sqrt{5} \), \( 64\sqrt{5} \). Wait, maybe I made a mistake in the segments. Wait, maybe \( QT = 10 \) and \( RT = 4 \), but the hypotenuse is not \( 14 \). Wait, no, the Geometric Mean (Leg) Theorem: in right triangle \( \triangle ABC \), right - angled at \( C \), with altitude \( CD \) to hypotenuse \( AB \), then \( AC^{2}=AD\times AB \) and \( BC^{2}=BD\times AB \). So, in our case, \( \triangle QSR \) is right - angled at \( S \), \( ST \) is altitude to \( QR \). So, \( SR^{2}=RT\times QR \), where \( QR = QT + RT \). \( QT = 10 \), \( RT = 4 \), so \( QR=14 \), then \( SR^{2}=4\times14 = 56 \), \( SR=\sqrt{56}=2\sqrt{14} \). Wait, but let's check again. Wait, maybe the length of \( QT \) is \( 10 \) and \( RT \) is \( 4 \), but maybe the triangle is different. Wait, no, the formula is clear. So, \( q^2=RT\times QR=4\times(10 + 4)=4\times14 = 56 \), so \( q=\sqrt{56}=2\sqrt{14} \). Wait, but let's check the options. The last option is \( 2\sqrt{14} \). Wait, but maybe I made a mistake in the segments. Wait, maybe \( QT = 10 \) and \( RT = 4 \), but the hypotenuse is \( QR = 10+4 = 14 \), and the leg \( q \) is \( SR \), so \( SR^{2}=RT\times QR=4\times14 = 56 \), so \( SR = \sqrt{56}=2\sqrt{14} \).

Wait, no, wait another approach. Maybe the two segments are \( QT = 10 \) and \( RT = 4 \), and the leg \( q \) is related to \( QT \) and \( RT \) as \( q=\sqrt{RT\times QT} \)? No, that's the geometric mean of the two segments, which is the altitude. Wait, the altitude \( s=\sqrt{QT\times RT}=\sqrt{10\times4}=\sqrt{40}=2\sqrt{10} \), but that's not our case. We need the leg. So, the leg \( q \): by Pythagoras, in \( \triangle STR \), \( q^{2}=s^{2}+4^{2} \), and in \( \triangle QTS \), \( r^{2}=s^{2}+10^{2} \), and in \( \triangle QSR \), \( (q^{2}+r^{2})=(10 + 4)^{2}=196 \). But also, by Geometric Mean (Leg) Theorem, \( q^{2}=4\times14 = 56 \), \( r^{2}=10\times14 = 140 \), and \( 56+140 = 196 \), which matches \( 14^{2}=196 \). So, \( q^{2}=56 \), \( q=\sqrt{56}=2\sqrt{14} \).

Wait, but let's check the options again. The options are \( 4\sqrt{5} \), \( 20\sqrt{5} \), \( 64\sqrt{5} \), \( 2\sqrt{14} \). So, the correct answer should be \( 2\sqrt{14} \). Wait, but maybe I misread the lengths. Wait, maybe \( QT = 10 \) and \( RT = 4 \), but the hypotenuse is not \( 14 \). Wait, no, the sum of the two segments of the hypotenuse is the hypotenuse. So, \( QR=QT + RT=10 + 4 = 14 \). So, the calculation is correct.

Wait, another way: Let's use the Geometric Mean (Leg) Theorem formula. The formula is \( \text{leg}^2=\text{adjacent segment}\times\text{hypotenuse} \). Here, the leg is \( q \), the adjacent segment is \( RT = 4 \), and the hypotenuse \( QR=QT + RT=10 + 4 = 14 \). So, \( q^{2}=4\times14 = 56 \), so \( q=\sqrt{56}=2\sqrt{14} \).

Answer:

\( 2\sqrt{14} \) (the last option: \( 2\sqrt{14} \))