Question

what proportion should you set up to check if the triangle are similar?
(there are two triangles with side lengths: one triangle has sides 28, 16, 24; the other has sides 12, 18, 21. and there are three proportion options:

1. \\(\frac{12}{16} = \frac{18}{24} = \frac{21}{28}\\)
2. \\(\frac{12}{28} = \frac{18}{16} = \frac{21}{24}\\)
3. \\(\frac{21}{28} = \frac{12}{24} = \frac{18}{16}\\))

Explanation:

Step1: Identify corresponding sides

For similar triangles, corresponding sides are proportional. Let's label the triangles: Triangle 1 (with sides 12, 18, 21) and Triangle 2 (with sides 16, 24, 28). We need to match the sides in order. So 12 corresponds to 16, 18 corresponds to 24, 21 corresponds to 28? Wait, no, wait. Wait the first triangle (let's say triangle ASB? Wait the labels: first triangle (lower left) has sides 12, 18, 21? Wait the image: the lower triangle (A, B, C?) has sides 12, 18, 21? Wait no, the left triangle (J, S, K) has sides 16, 24, 28. The lower triangle (A, B, C) has sides 12, 18, 21. So we need to check the ratios. Let's list the sides:

Triangle 1 (A, B, C?): sides 12, 18, 21

Triangle 2 (J, S, K): sides 16, 24, 28

Now, let's find the ratios:

12/16 = 3/4, 18/24 = 3/4, 21/28 = 3/4. So the proportion should be 12/16 = 18/24 = 21/28. Wait, looking at the options, the first option (top blue box) is 12/16 = 18/24 = 21/28. Let's check the options:

First option: 12/16 = 18/24 = 21/28

Second option: 12/28 = 18/16 = 21/24 (ratios not equal)

Third option: 21/28 = 12/24 = 18/16 (ratios not equal)

So the correct proportion is 12/16 = 18/24 = 21/28, which is the first option.

Step2: Verify the ratios

Calculate each ratio:

12/16 = 3/4, 18/24 = 3/4, 21/28 = 3/4. All equal, so this proportion is correct.

Answer:

The correct proportion is the top blue box: $\frac{12}{16} = \frac{18}{24} = \frac{21}{28}$ (the first option in the image, likely labeled as the top one with the three equal ratios 12/16, 18/24, 21/28)