Question

how can you prove that the triangles are similar? (1 point) aa - sas - sss - the triangles are not similar.

Explanation:

Step1: Identify corresponding sides

Label the sides: For $\triangle ABC$, sides are $AB = 39$, $BC = 32$, $AC = 60$. For $\triangle DEF$, sides are $DF = 13$, $EF = 8$, $DE = 15$.

Step2: Calculate ratios of corresponding sides

• Ratio of $AB$ to $DF$: $\frac{AB}{DF}=\frac{39}{13} = 3$
• Ratio of $BC$ to $EF$: $\frac{BC}{EF}=\frac{32}{8}=4$
• Ratio of $AC$ to $DE$: $\frac{AC}{DE}=\frac{60}{15} = 4$

Wait, correction: Wait, maybe I mixed up corresponding sides. Let's re - check. Let's pair the longest sides, medium, and shortest.

Longest side of $\triangle ABC$: $AC = 60$; Longest side of $\triangle DEF$: $DE = 15$. Ratio: $\frac{60}{15}=4$.

Medium side of $\triangle ABC$: $AB = 39$; Medium side of $\triangle DEF$: $DF = 13$. Ratio: $\frac{39}{13}=3$.

Shortest side of $\triangle ABC$: $BC = 32$; Shortest side of $\triangle DEF$: $EF = 8$. Ratio: $\frac{32}{8}=4$.

Wait, the ratios are not equal. Wait, maybe I paired the wrong sides. Let's check the angles. Wait, no, the SSS similarity requires all three sides to be in proportion. Wait, maybe I made a mistake in side pairing.

Wait, let's list the sides in order (ascending) for each triangle.

For $\triangle ABC$: $BC = 32$, $AB = 39$, $AC = 60$ (sorted: $32, 39, 60$)

For $\triangle DEF$: $EF = 8$, $DF = 13$, $DE = 15$ (sorted: $8, 13, 15$)

Now, check ratios:

$\frac{32}{8}=4$, $\frac{39}{13}=3$, $\frac{60}{15}=4$. Since the ratios are not all equal (3 vs 4), wait, that can't be. Wait, maybe the sides are $AB = 39$, $BC = 32$, $AC = 60$ and $DF = 13$, $EF = 8$, $DE = 15$. Wait, maybe the included angle? Wait, no, let's re - calculate.

Wait, $\frac{AB}{DF}=\frac{39}{13}=3$, $\frac{BC}{EF}=\frac{32}{8}=4$, $\frac{AC}{DE}=\frac{60}{15}=4$. Wait, no, if we take $BC$ and $EF$ as one pair, $AC$ and $DE$ as another, and $AB$ and $DF$ as another, the ratios are not the same. Wait, maybe the triangles are not similar? But that's an option. Wait, let's recalculate:

Wait, $39\div13 = 3$, $32\div8 = 4$, $60\div15 = 4$. Since the ratios are not all equal (3 ≠ 4), the SSS similarity condition is not met. Also, AA: we don't know angles. SAS: we don't know included angles. So the triangles are not similar? Wait, but let's check again.

Wait, maybe I mixed up the sides. Let's see: $AB = 39$, $AC = 60$, $BC = 32$; $DF = 13$, $DE = 15$, $EF = 8$.

Check $\frac{AB}{DF}=\frac{39}{13}=3$, $\frac{AC}{DE}=\frac{60}{15}=4$, $\frac{BC}{EF}=\frac{32}{8}=4$. So two ratios are 4, one is 3. So not all equal. So the triangles are not similar? But wait, maybe I made a mistake. Wait, let's check the problem again.

Wait, the options are AA, SAS, SSS, or not similar. Let's check the ratios again.

Wait, $39/13 = 3$, $32/8 = 4$, $60/15 = 4$. So the ratios of the sides are not proportional (since 3 ≠ 4). So the triangles are not similar.

Wait, but maybe I paired the wrong sides. Let's try another pairing. Suppose $AB$ corresponds to $DE$, $BC$ to $EF$, $AC$ to $DF$.

$\frac{AB}{DE}=\frac{39}{15}=\frac{13}{5}$, $\frac{BC}{EF}=\frac{32}{8}=4$, $\frac{AC}{DF}=\frac{60}{13}\approx4.615$. Not equal.

Another pairing: $AB$ to $EF$, $BC$ to $DF$, $AC$ to $DE$.

$\frac{39}{8}=4.875$, $\frac{32}{13}\approx2.46$, $\frac{60}{15}=4$. Not equal.

So since the ratios of the corresponding sides are not equal, the triangles are not similar.

Answer:

The triangles are not similar.