Question

the right triangle on the right is a scaled copy of the right triangle on the left. identify the scale factor. express your answer as a whole number or fraction in simplest form.

10

5

24

12

Explanation:

Step1: Recall scale factor formula

Scale factor \( = \frac{\text{Length of corresponding side in image}}{\text{Length of corresponding side in original}} \)

Step2: Identify corresponding sides

Left triangle (original) has legs 5, 12. Right triangle (image) has legs 24, 10. Wait, 5 and 24? No, wait, maybe I mixed up. Wait, no—wait, the left triangle: vertical leg 5, horizontal leg 12. Right triangle: vertical leg 24, horizontal leg 10? Wait, no, maybe the vertical legs: left is 5, right is 24? No, that can't be. Wait, no, maybe the horizontal and vertical. Wait, no, maybe I got the correspondence wrong. Wait, wait, the left triangle: vertical side 5, horizontal side 12. The right triangle: vertical side 24, horizontal side 10? No, that doesn't make sense. Wait, no—wait, maybe the hypotenuse? No, wait, no, let's check the legs. Wait, maybe the left triangle's vertical leg is 5, and the right triangle's vertical leg is 24? No, that would be scale factor 24/5, but the horizontal sides: left is 12, right is 10? No, that's not proportional. Wait, I must have misidentified the corresponding sides. Wait, no—wait, the left triangle: vertical leg 5, horizontal leg 12. The right triangle: vertical leg 24, horizontal leg 10? No, that's not. Wait, maybe the right triangle is rotated. So the left triangle: vertical leg 5, horizontal leg 12. The right triangle: vertical leg 24 (which would correspond to the horizontal leg of the left? No, 12 and 24? Wait, 122=24? But 5?=10? Wait, 52=10. Oh! Wait, I see. The left triangle: vertical leg 5, horizontal leg 12. The right triangle: vertical leg 24? No, wait, no—wait, the right triangle's horizontal side is 10, and vertical side is 24? Wait, no, maybe I flipped the horizontal and vertical. Wait, the left triangle: vertical leg 5, horizontal leg 12. The right triangle: horizontal leg 10, vertical leg 24? No, that's not. Wait, no—wait, the right triangle is a scaled copy, so corresponding sides must be proportional. So let's check: left triangle has sides 5, 12, and hypotenuse (5²+12²=25+144=169, so hypotenuse 13). Right triangle: sides 10, 24, and hypotenuse (10²+24²=100+576=676, which is 26²). Wait, 52=10, 122=24, 132=26. Oh! Wait, I misread the right triangle's vertical side. Wait, the right triangle's vertical side is 24? No, wait, no—wait, the left triangle: vertical leg 5, horizontal leg 12, hypotenuse 13. The right triangle: horizontal leg 10, vertical leg 24? No, that's not. Wait, no, the right triangle's vertical leg is 24? Wait, no, I think I made a mistake in the diagram. Wait, the left triangle: vertical side 5, horizontal side 12. The right triangle: vertical side 24? No, wait, the right triangle's horizontal side is 10, and vertical side is 24? No, that can't be. Wait, no—wait, the right triangle is rotated, so the vertical leg of the left (5) corresponds to the horizontal leg of the right (10), and the horizontal leg of the left (12) corresponds to the vertical leg of the right (24). Let's check: 10/5 = 2, 24/12 = 2. Ah! There we go. So the right triangle is a scaled copy where the left triangle's vertical leg (5) corresponds to the right triangle's horizontal leg (10), and left's horizontal leg (12) corresponds to right's vertical leg (24). So scale factor is 10/5 = 2, or 24/12 = 2. So scale factor is 2? Wait, no—wait, 10/5 is 2, 24/12 is 2. So yes, scale factor is 2. Wait, but let's confirm. So original (left) has side 5, image (right) has corresponding side 10: 10/5 = 2. Original side 12, image side 24: 24/12 = 2. So scale factor is 2.

Answer:

\( 2 \)