Question

1. which triangle is similar to tuv?

Explanation:

Step1: Find ratios of TUV's sides

Triangle TUV has sides \( VT = 42 \), \( TU = 70 \), \( UV = 84 \). Simplify ratios:
\( \frac{42}{70}=\frac{3}{5} \), \( \frac{42}{84}=\frac{1}{2} \), \( \frac{70}{84}=\frac{5}{6} \)? Wait, better to divide all by GCD(42,70,84)=14:
\( 42\div14 = 3 \), \( 70\div14 = 5 \), \( 84\div14 = 6 \). So sides ratio \( 3:5:6 \).

Step2: Check Option A (FGH: 8,12,14)

Simplify by GCD(8,12,14)=2: \( 4,6,7 \). Ratio \( 4:6:7
eq 3:5:6 \).

Step3: Check Option B (ABC: 48,72,84)

GCD(48,72,84)=12: \( 48\div12 = 4 \)? Wait, no: 48÷12=4? Wait 48÷12=4, 72÷12=6, 84÷12=7? No, wait 48,72,84: divide by 12? Wait 48÷12=4, 72÷12=6, 84÷12=7. Wait no, earlier TUV was 3:5:6. Wait, wait I miscalculated TUV. Wait TUV: VT=42, TU=70, UV=84. So 42:70:84. Divide by 14: 3:5:6. Now ABC: 48,72,84. Wait 48:72:84. Divide by 12: 4:6:7? No, wait 48÷12=4, 72÷12=6, 84÷12=7. No. Wait wait, maybe I mixed up sides. Wait TUV: VT=42 (vertical), TU=70 (horizontal), UV=84 (hypotenuse). ABC: CB=48, AB=72, AC=84. So CB=48, AB=72, AC=84. Let's find ratios: 48:72:84. Divide by 12: 4:6:7? No, wait 48÷24=2, 72÷24=3, 84÷24=3.5? No, wait TUV: 42,70,84. 42/70=0.6, 42/84=0.5, 70/84≈0.833. ABC: 48/72≈0.666, 48/84≈0.571, 72/84≈0.857. Wait no, earlier mistake: TUV sides: VT=42, TU=70, UV=84. So sides are 42 (leg), 70 (leg), 84 (hypotenuse). So legs 42 and 70, hypotenuse 84. For ABC: CB=48 (leg), AB=72 (leg), AC=84 (hypotenuse). Check leg ratios: 42/70 = 0.6; 48/72 = 0.666… No. Wait wait, maybe I flipped. Wait TUV: VT=42, TU=70, UV=84. So 42 (VT), 70 (TU), 84 (UV). So 42:70:84 = 3:5:6 (divided by 14). Now ABC: CB=48, AB=72, AC=84. 48:72:84. Divide by 12: 4:6:7. No. Wait Option C: SQR: 15,25,30. Wait 15:25:30. Divide by 5: 3:5:6. Oh! Wait the third triangle (C) has sides 15,25,30? Wait the image: SQR: QS=15, QR=25, SR=30? Wait the user's image: third triangle (C) has 15,25,30. Wait I missed that. Wait original problem: the third triangle (C) has sides 15,25,30? Wait the user's diagram: S, Q, R: QS=15, QR=25, SR=30? Wait let's recheck.

Wait the user's problem: Triangle TUV: 42,70,84. Then Option A: 8,12,14. Option B: 48,72,84. Option C: 15,25,30? Wait the user's image: third triangle (C) has 15,25,30. Let's check that. 15:25:30. Divide by 5: 3:5:6. Which matches TUV's 3:5:6 (42,70,84 divided by 14 is 3,5,6). So I must have misread Option C. Wait the user's diagram: third triangle (C) has sides 15,25,30. So:

TUV: 42,70,84 → 3:5:6 (÷14)
Option C: 15,25,30 → 3:5:6 (÷5)

Ah! I missed Option C earlier. Wait the user's options: A, B, and C (SQR: 15,25,30). So the correct triangle is the one with sides 15,25,30 (ratio 3:5:6), which matches TUV's ratio. But in the user's options, maybe the third one is C. Wait the user's problem: the three options are A, B, C. Let's re-express:

TUV sides: 42 (VT), 70 (TU), 84 (UV) → 42/70 = 0.6, 42/84 = 0.5, 70/84 ≈0.833.
Triangle C (SQR): 15 (QS), 25 (QR), 30 (SR) → 15/25=0.6, 15/30=0.5, 25/30≈0.833. Perfect match. So the similar triangle is the one with sides 15,25,30 (Option C). Wait the user's options: A (8,12,14), B (48,72,84), C (15,25,30). So the correct answer is the triangle with sides 15,25,30 (ratio 3:5:6), which is similar to TUV (ratio 3:5:6).

Answer:

The triangle with sides 15, 25, 30 (the third triangle, likely labeled as C) is similar to TUV. If the options are A, B, C as per the diagram, and C has sides 15,25,30, then the answer is C (the triangle with sides 15, 25, 30).