Question

if these two shapes are similar, what is the measure of the missing length u? 4 yd 8 yd 13 yd u = \boxed{} yards submit

Explanation:

Step1: Identify the ratio of corresponding sides

For similar triangles, the ratios of corresponding sides are equal. The sides of the smaller triangle are 4 yd and 8 yd, and the corresponding side of the larger triangle to 4 yd is 13 yd? Wait, no, wait. Wait, the smaller triangle has sides 4 yd and 8 yd, and the larger triangle has a side 13 yd (corresponding to the 4 yd side?) Wait, no, let's check the correspondence. Wait, the smaller triangle: one side is 4 yd, another is 8 yd. The larger triangle: one side is 13 yd (corresponding to 4 yd?), and the other is u (corresponding to 8 yd?). Wait, no, maybe the ratio is 8/4 = 2? Wait, no, wait. Wait, the smaller triangle: 4 yd and 8 yd. The larger triangle: 13 yd and u. Wait, no, maybe the corresponding sides are 4 yd (small) and 13 yd (large)? No, that can't be. Wait, no, the similar triangles: the ratio of corresponding sides. Let's see, the smaller triangle has sides 4 and 8, so the ratio of the larger to smaller is (let's say the side of length 8 in small corresponds to u in large, and side 4 in small corresponds to 13 in large? Wait, no, that would be 13/4 = u/8. Let's check:

If the two triangles are similar, then the ratio of corresponding sides is equal. So, let's assume that the side of length 4 yd in the smaller triangle corresponds to the side of length 13 yd in the larger triangle, and the side of length 8 yd in the smaller triangle corresponds to the side of length u in the larger triangle. Then the ratio is 13/4 = u/8.

Step2: Solve for u

Cross-multiplying: 4u = 13 * 8
4u = 104
Divide both sides by 4: u = 104 / 4 = 26? Wait, no, that can't be. Wait, maybe I got the correspondence wrong. Wait, maybe the smaller triangle's 8 yd corresponds to the larger triangle's 13 yd? No, that would be 13/8 = u/4. Then u = (134)/8 = 52/8 = 6.5. No, that doesn't make sense. Wait, wait, maybe the sides are 4 and 8 (small), and 13 and u (large), with 4 corresponding to 13? No, that would be 13/4 = u/8. Then u = (138)/4 = 26. Wait, but let's check the ratio. The smaller triangle: 4 and 8, so 8 is twice 4. So the larger triangle should have u twice 13? Wait, 132=26. Yes, that makes sense. Because 8/4 = 2, so the scale factor is 2. So the side corresponding to 13 yd (which is 4 yd scaled by 2? No, wait, 42=8, but 132=26. Wait, maybe the 4 yd side in small corresponds to 13 yd in large? No, that would mean scale factor 13/4, which is not 2. Wait, I think I messed up the correspondence. Let's look at the triangles. The smaller triangle: one side is 4, another is 8. The larger triangle: one side is 13, another is u. If the triangles are similar, the ratio of the sides should be equal. So 8/4 = u/13? Wait, 8/4 is 2, so u = 132 = 26. Yes, that makes sense. Because 4 and 13? No, wait, 4 (small) and 13 (large)? No, 8 (small) and u (large), 4 (small) and 13 (large). Wait, no, 4 and 13: 13/4, 8 and u: u/8. If they are equal, 13/4 = u/8 → u = (138)/4 = 26. Yes, that's correct. So the scale factor is 2, because 8/4 = 2, so u = 132 = 26.

Answer:

26