Question

Are these two triangles similar. If so why, make sure to explain your answer and what similarity theorem you can
use (AA, SAS, SSS~). If similar also create a similarity statement. (KLM triangle. KL is 20, LM is 15, MK is 10. QRP triangle. QR is 24, RP is 12, PQ is 18)

Answer 1

To determine whether triangles KLM and QRP are similar, we can use the Side-Side-Side (SSS) similarity theorem. Two triangles are similar by the SSS similarity theorem if the ratios of the corresponding sides are equal.

First, let's list the side lengths for both triangles:

**Triangle KLM:**
- KL = 20
- LM = 15
- MK = 10

**Triangle QRP:**
- QR = 24
- RP = 12
- PQ = 18

Next, we will find the ratios of the corresponding sides of the triangles.

1. **Ratio of KL to QR:**
\[
\frac{KL}{QR} = \frac{20}{24} = \frac{5}{6}
\]

2. **Ratio of LM to RP:**
\[
\frac{LM}{RP} = \frac{15}{12} = \frac{5}{4}
\]

3. **Ratio of MK to PQ:**
\[
\frac{MK}{PQ} = \frac{10}{18} = \frac{5}{9}
\]

Now let's compare the ratios:

- The ratio of KL to QR is \(\frac{5}{6}\)
- The ratio of LM to RP is \(\frac{5}{4}\)
- The ratio of MK to PQ is \(\frac{5}{9}\)

Since these ratios (\(\frac{5}{6}\), \(\frac{5}{4}\), and \(\frac{5}{9}\)) are not all equal, we can conclude that the sides of the triangles are not in proportion.

Thus, triangles KLM and QRP are **not similar** based on the SSS similarity criterion. Therefore, there is no similarity statement to create for these triangles.