Question

question 7
10 pts
g.gsr.5.3 (mc)
determine if the pair of triangles are similar. if similar, choose the correct postulate or theorem to prove similarity.
image of two triangles with side lengths: left triangle 24, 30, 36; right triangle 21, 25, 30
accessible description
options:

• not similar
• similar; aa similarity postulate
• similar; sas similarity postulate
• similar; sss similarity postulate

Explanation:

Step1: Identify triangle sides

First triangle sides: \(24\), \(30\), \(36\)
Second triangle sides: \(21\), \(25\), \(30\)

Step2: Check SSS similarity (ratios)

Calculate ratios of corresponding sides:
\(\frac{24}{21} = \frac{8}{7}\), \(\frac{30}{25} = \frac{6}{5}\), \(\frac{36}{30} = \frac{6}{5}\)

Ratios are not equal (\(\frac{8}{7}
eq \frac{6}{5}\)), so SSS fails.

Wait, recheck sides: Maybe I mixed up. Let's list sides in order (smallest to largest):

First triangle: \(24\), \(30\), \(36\) (sorted: \(24, 30, 36\))
Second triangle: \(21\), \(25\), \(30\) (sorted: \(21, 25, 30\))

Wait, no—maybe the sides are: First triangle: \(24\) (height), \(30\) (base), \(36\) (hypotenuse). Second: \(21\) (height), \(25\) (base? No, 25, 21, 30). Wait, 30 is common? Wait, 30 in first triangle (base) and 30 in second (one side). Wait, maybe I misread. Let's check again:

First triangle: sides \(24\), \(30\), \(36\)
Second triangle: sides \(21\), \(25\), \(30\)

Wait, no—maybe the sides are: First: \(24\), \(30\), \(36\); Second: \(21\), \(30\), \(25\)? No, the second triangle has \(25\), \(21\), \(30\). Let's check ratios of \(24/21\), \(30/25\), \(36/30\):

\(24/21 = 8/7 \approx 1.14\)
\(30/25 = 6/5 = 1.2\)
\(36/30 = 6/5 = 1.2\)

Wait, two ratios are \(6/5\), one is \(8/7\). Not all equal. But wait, maybe the sides are paired differently. Wait, maybe the first triangle's sides are \(24\), \(30\), \(36\); second: \(21\), \(25\), \(30\). Wait, no—maybe the right angle? Wait, are these right triangles? Let's check Pythagoras:

First triangle: \(24^2 + 30^2 = 576 + 900 = 1476\); \(36^2 = 1296\). Not right. Second: \(21^2 + 25^2 = 441 + 625 = 1066\); \(30^2 = 900\). Not right. So not right triangles.

Wait, maybe I made a mistake. Let's re-express the sides:

First triangle: \(24\), \(30\), \(36\)
Second triangle: \(21\), \(25\), \(30\)

Wait, 30 is a side in both. Let's check \(24/21 = 8/7\), \(30/30 = 1\), \(36/25 = 36/25\). No, that's worse.

Wait, maybe the sides are: First triangle: \(24\), \(30\), \(36\); Second: \(21\), \(30\), \(25\)? No, the second triangle's sides are \(25\), \(21\), \(30\). Wait, maybe the problem is that the triangles are not similar? But let's check again.

Wait, maybe I mixed up the sides. Let's list the sides of each triangle:

Triangle 1: \(24\), \(30\), \(36\)
Triangle 2: \(21\), \(25\), \(30\)

Now, let's find the ratios of corresponding sides (matching the order of length):

Triangle 1 sorted: \(24\), \(30\), \(36\)
Triangle 2 sorted: \(21\), \(25\), \(30\)

So ratio of smallest: \(24/21 = 8/7\)
Ratio of middle: \(30/25 = 6/5\)
Ratio of largest: \(36/30 = 6/5\)

Since the ratios of the middle and largest sides are equal (\(6/5\)), but the smallest is different (\(8/7
eq 6/5\)), the sides are not proportional. Therefore, the triangles are not similar. Wait, but the options include "Not Similar". Let's confirm:

For SSS similarity, all three ratios must be equal. Here, \(24/21 = 8/7\), \(30/25 = 6/5\), \(36/30 = 6/5\). Since \(8/7
eq 6/5\), the ratios are not all equal. Therefore, the triangles are not similar.

Answer:

Not Similar