The graph below shows triangle A and triangle B. The side lengths of triangle A are proportional to the side lengths of triangle B, and the corresponding angle measures of the two triangles are equal.

Question

Triangle A has vertices at (0, 3), (2, 4) and (3, 1). Triangle B has vertices at (0, 6), (4, 8) and (6, 2).

Which statement BEST describes the relationship between triangle A and triangle B?

A
.

The triangles are similar but not congruent.
B
.

The triangles are congruent but not similar.
C.

The triangles are both similar and congruent.
D.

The triangles are neither similar nor congruent.

Answer 1

To determine the relationship between triangle A and triangle B, we can analyze the side lengths and angles of both triangles.

First, let's find the lengths of the sides of triangle A. The vertices of triangle A are at (0, 3), (2, 4), and (3, 1).

1. Length of side between (0, 3) and (2, 4): \[ \sqrt{(2 - 0)^2 + (4 - 3)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]

2. Length of side between (2, 4) and (3, 1): \[ \sqrt{(3 - 2)^2 + (1 - 4)^2} = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]

3. Length of side between (3, 1) and (0, 3): \[ \sqrt{(0 - 3)^2 + (3 - 1)^2} = \sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]

So, the side lengths of triangle A are:

• \( \sqrt{5} \)
• \( \sqrt{10} \)
• \( \sqrt{13} \)

Next, let's find the lengths of the sides of triangle B, which has vertices at (0, 6), (4, 8), and (6, 2).

1. Length of side between (0, 6) and (4, 8): \[ \sqrt{(4 - 0)^2 + (8 - 6)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} \]

2. Length of side between (4, 8) and (6, 2): \[ \sqrt{(6 - 4)^2 + (2 - 8)^2} = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} \]

3. Length of side between (6, 2) and (0, 6): \[ \sqrt{(0 - 6)^2 + (6 - 2)^2} = \sqrt{(-6)^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} \]

So, the side lengths of triangle B are:

• \( \sqrt{20} \)
• \( \sqrt{40} \)
• \( \sqrt{52} \)

Now, let's determine the proportionality of the side lengths.

We can express the side lengths of triangle B in terms of triangle A:

• \( \sqrt{20} = 2\sqrt{5} \)
• \( \sqrt{40} = 2\sqrt{10} \)
• \( \sqrt{52} = 2\sqrt{13} \)

Therefore, each side length of triangle A is multiplied by 2 to get the corresponding side length in triangle B.

This confirms the triangles are similar because:

1. The side lengths are in proportion (there is a consistent ratio).
2. The corresponding angles are equal.

Since triangle A and triangle B have proportional side lengths and equal angles, they are similar but not congruent (as the corresponding side lengths are not equal).

Therefore, the best statement that describes the relationship between triangle A and triangle B is:

A. The triangles are similar but not congruent.