Two triangles are graphed on a coordinate plane. Triangle MNP has vertices `M(-4, 2), N(-4, 6), and P(-6, 2).` Triangle QRS` has vertices `Q(-5, -1), R=(-5, -5), and S(4, -5)`. Which of the following statements is true?

Options

A. `triangleMNP` is congruent to `triangleQRS`
B. `triangleMNP` is similar to `triangleQRS`
C. `triangleMNP` is similar to `triangleRSQ`
D. The triangle are neither congruent nor similar.

Answer

To determine the relationship between the two triangles MNP and QRS based on their vertices, we will follow these steps:

### Step 1: Plot the Points
We will plot the vertices of both triangles on a coordinate plane.

- Triangle MNP has vertices:
- M(-4, 2)
- N(-4, 6)
- P(-6, 2)

- Triangle QRS has vertices:
- Q(-5, -1)
- R(-5, -5)
- S(4, -5)

### Step 2: Calculate the Lengths of the Sides
Next, we will calculate the lengths of the sides of both triangles.

**For Triangle MNP:**
- Length of MN:
\[ MN = |y_2 - y_1| = |6 - 2| = 4 \]

- Length of MP:
\[ MP = |x_2 - x_1| = |-6 - (-4)| = 2 \]

- Length of NP:
\[ NP = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-4 - (-6))^2 + (6 - 2)^2} = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

**For Triangle QRS:**
- Length of QR:
\[ QR = |y_2 - y_1| = |-5 - (-1)| = 4 \]

- Length of QS:
\[ QS = |x_2 - x_1| = |4 - (-5)| = 9 \]

- Length of RS:
\[ RS = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-5 - 4)^2 + (-5 - (-5))^2} = \sqrt{(-9)^2 + (0)^2} = 9 \]

### Step 3: Compare the Ratios of the Sides
Now, we will compare the ratios of the corresponding sides of both triangles.

- Ratios for Triangle MNP:
- MN: 4
- MP: 2
- NP: \(2\sqrt{5}\)

- Ratios for Triangle QRS:
- QR: 4
- QS: 9
- RS: 9

### Step 4: Check for Similarity
To check for similarity, we need to see if the ratios of the corresponding sides are equal.

- Comparing MN and QR:
\[ \frac{MN}{QR} = \frac{4}{4} = 1 \]

- Comparing MP and QS:
\[ \frac{MP}{QS} = \frac{2}{9} \]

- Comparing NP and RS:
\[ \frac{NP}{RS} = \frac{2\sqrt{5}}{9} \]

Since the ratios are not equal, the triangles are not similar.

### Step 5: Check for Congruence
For the triangles to be congruent, all corresponding sides must be equal. Since we have already found that the lengths of the sides are different, the triangles are not congruent.

### Conclusion
Both triangles MNP and QRS are neither similar nor congruent.

### Final Answer
The correct statement is that both triangles are not congruent and not similar.

---