Question

Using any of the three triangle similarities of corresponding angles or side lengths, explain why the two triangles are similar. Reference the triangle similarity criteria of choice (SAS or SSS) explain your reasoning in your answer, and show work for any needed computations.

A(-3,-2)
B(4,-2)
C(-3,3)

A'(-6,-2)
B'(8,-2)
C'(-6,8)

Answer 1

To determine if triangles ABC and A'B'C' are similar, we can use the Side-Side-Side (SSS) similarity criterion, which states that if the corresponding sides of two triangles are in proportion, then the triangles are similar.

### Step 1: Calculate the lengths of the sides of triangle ABC

1. **Length of side AB**:
\[
AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(4 - (-3))^2 + (-2 - (-2))^2} = \sqrt{(4 + 3)^2 + (0)^2} = \sqrt{7^2} = 7
\]

2. **Length of side BC**:
\[
BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(-3 - 4)^2 + (3 - (-2))^2} = \sqrt{(-7)^2 + (5)^2} = \sqrt{49 + 25} = \sqrt{74}
\]

3. **Length of side CA**:
\[
CA = \sqrt{(x_A - x_C)^2 + (y_A - y_C)^2} = \sqrt{(-3 - (-3))^2 + (-2 - 3)^2} = \sqrt{(0)^2 + (-5)^2} = \sqrt{25} = 5
\]

### Step 2: Calculate the lengths of the sides of triangle A'B'C'

1. **Length of side A'B'**:
\[
A'B' = \sqrt{(x_{B'} - x_{A'})^2 + (y_{B'} - y_{A'})^2} = \sqrt{(8 - (-6))^2 + (-2 - (-2))^2} = \sqrt{(8 + 6)^2 + (0)^2} = \sqrt{14^2} = 14
\]

2. **Length of side B'C'**:
\[
B'C' = \sqrt{(x_{C'} - x_{B'})^2 + (y_{C'} - y_{B'})^2} = \sqrt{(-6 - 8)^2 + (8 - (-2))^2} = \sqrt{(-14)^2 + (10)^2} = \sqrt{196 + 100} = \sqrt{296}
\]

3. **Length of side C'A'**:
\[
C'A' = \sqrt{(x_{A'} - x_{C'})^2 + (y_{A'} - y_{C'})^2} = \sqrt{(-6 - (-6))^2 + (-2 - 8)^2} = \sqrt{(0)^2 + (-10)^2} = \sqrt{100} = 10
\]

### Step 3: Proportions of the sides

Now, we compare the lengths of the sides of triangles ABC and A'B'C':

- Side lengths of triangle ABC are \( AB = 7, BC = \sqrt{74}, CA = 5 \).
- Side lengths of triangle A'B'C' are \( A'B' = 14, B'C' = \sqrt{296}, C'A' = 10 \).

To check for proportionality:

1. **Ratio of AB to A'B'**:
\[
\frac{AB}{A'B'} = \frac{7}{14} = \frac{1}{2}
\]

2. **Ratio of CA to C'A'**:
\[
\frac{CA}{C'A'} = \frac{5}{10} = \frac{1}{2}
\]

3. **Ratio of BC to B'C'**:
\[
\frac{BC}{B'C'} = \frac{\sqrt{74}}{\sqrt{296}} = \frac{\sqrt{74}}{\sqrt{4 \times 74}} = \frac{\sqrt{74}}{2\sqrt{74}} = \frac{1}{2}
\]

### Conclusion

Since the ratios of the corresponding sides of triangles ABC and A'B'C' are equal (\(\frac{1}{2}\)), by the SSS (Side-Side-Side) criterion, we can conclude that triangles ABC and A'B'C' are similar.

Thus, **triangles ABC and A'B'C' are similar by the SSS similarity criterion.**