Question

1. q is equidistant from the sides of ∠tsr. find the value of x. (2x + 24)° 30° options: 27, 15, 30, 3

Explanation:

Step1: Recall Angle Bisector Theorem

A point equidistant from the sides of an angle lies on the angle bisector. So, \( SQ \) bisects \( \angle TSR \), meaning \( \angle TSQ=\angle RSQ = 30^\circ \), and \( \angle TSR=2\times30^\circ=(2x + 24)^\circ \).

Step2: Set Up Equation

Set \( 2\times30=2x + 24 \). Simplify left side: \( 60 = 2x+24 \).

Step3: Solve for x

Subtract 24 from both sides: \( 60 - 24=2x \), so \( 36 = 2x \). Divide by 2: \( x = 18 \)? Wait, no, wait. Wait, the angle \( \angle TSR \) is \( (2x + 24)^\circ \), and since \( SQ \) is the bisector, each part is \( 30^\circ \), so \( \angle TSR=2\times30^\circ = 60^\circ \), so \( 2x + 24=60 \). Then \( 2x=60 - 24=36 \), so \( x = 18 \)? But the options are 27,15,30,3. Wait, maybe I misread. Wait, the angle at S: the total angle is \( (2x + 24)^\circ \), and the bisector splits it into two \( 30^\circ \) angles? Wait, no, maybe the angle \( (2x + 24)^\circ \) is the total, and the bisector makes each part \( 30^\circ \), so \( 2x + 24=2\times30 \)? Wait, no, maybe the angle \( (2x + 24)^\circ \) is equal to \( 2\times30^\circ \)? Wait, no, let's re - examine the diagram. The right angles: QT is perpendicular to ST, QR is perpendicular to SR, and QT = QR (marked with ticks), so SQ is the angle bisector. So \( \angle QST=\angle QSR = 30^\circ \), so \( \angle TSR=\angle QST+\angle QSR=30^\circ + 30^\circ = 60^\circ \). So \( 2x + 24=60 \). Then \( 2x=60 - 24 = 36 \), \( x = 18 \). But 18 is not in the options. Wait, maybe the angle \( (2x + 24)^\circ \) is one of the bisected angles? Wait, maybe I got the angle wrong. Let's look again. The diagram: the angle at S has a curved arrow with \( (2x + 24)^\circ \), and a 30° angle. If SQ is the bisector, then \( (2x + 24)^\circ=30^\circ\times2 \)? No, wait, maybe the angle \( (2x + 24)^\circ \) is equal to \( 30^\circ\times2 \)? Wait, no, maybe the equation is \( 2x + 24=2\times30 \)? Wait, 2×30 = 60, 2x+24 = 60, 2x=36, x = 18. But the options are 27,15,30,3. Wait, maybe the angle \( (2x + 24)^\circ \) is the angle of the bisector? No, that doesn't make sense. Wait, maybe I misread the angle. Wait, the angle with the curved arrow is \( (2x + 24)^\circ \), and the other angle is 30°, and SQ is the bisector, so \( 2x + 24=30\times2 \)? No, 30×2=60, 2x + 24=60, x=18. But the options don't have 18. Wait, maybe the angle is \( (2x + 24)^\circ \) and the bisected angle is \( 30^\circ \), so \( 2x + 24=30 \)? No, that would be negative. Wait, maybe the total angle is \( (2x + 24)^\circ \), and the bisector makes each part \( 30^\circ \), so \( 2x + 24=2\times30 \), but the options are wrong? No, maybe I made a mistake. Wait, let's check the options again. The options are 27,15,30,3. Let's try x = 3: 2×3+24=30, which is equal to 2×15? No. Wait, x=3: 2x +24=30, and if the angle is 30°, but the bisected angle would be 15°, no. Wait, x=15: 2×15 +24=54, no. x=27: 2×27+24=78, no. x=3: 30, no. Wait, maybe the angle \( (2x + 24)^\circ \) is equal to \( 30^\circ \times2 \)? No, 60. Wait, maybe the problem is that the angle \( (2x + 24)^\circ \) is the same as \( 2\times30^\circ \), but the options don't have 18. Wait, maybe I misread the diagram. Wait, the right angles: QT ⊥ ST, QR ⊥ SR, and QT = QR (the ticks), so SQ is the angle bisector. So \( \angle QST=\angle QSR \). The angle \( \angle QST \) is \( 30^\circ \), so \( \angle QSR=30^\circ \), so \( \angle TSR=\angle QST+\angle QSR = 60^\circ \), so \( 2x + 24=60 \), x=18. But since 18 is not an option, maybe there's a misprint, but among the options, maybe I made a mistake. Wait, maybe the angle \( (2x + 24)^\circ \) is equal to \( 30^\circ \), no. Wait, maybe the angle is \( (2x + 24)^\circ \) and the bisected angle is \( 30^\circ \), so \( 2x + 24=30 \), x=3. Ah! Maybe I misread the diagram. Maybe the angle \( (2x + 24)^\circ \) is one of the bisected angles, not the total. So if SQ is the bisector, then \( 2x + 24=30 \), so \( 2x=6 \), x=3. That's one of the options (3). So probably, the angle \( (2x + 24)^\circ \) is equal to \( 30^\circ \), so solving \( 2x + 24=30 \), we get \( 2x=6 \), \( x = 3 \).

Answer:

3