https://file.aiquickdraw.com/mathbot/file/1727357354395hrixo9wr.jpg

The problem you've shared is asking for the following:

### Given:
In a coordinate system \( Oxy \), with a unit scale where 1 cm represents 1 unit, the points \( M \), \( N \), and \( P \) are provided.

### Questions:
1. **A)** Write down the coordinates of points \( M \), \( N \), and \( P \).
2. **B)** Write down the coordinates of point \( Q \), which is symmetric to point \( N \) with respect to the origin.
3. **C)** Find the area of triangle \( \Delta MNP \).

### Let's solve these step by step:

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### A) Coordinates of points \( M \), \( N \), and \( P \):

From the grid in the image, it looks like the coordinates are:

- \( M(-4, 3) \)
- \( N(1, -2) \)
- \( P(4, 2) \)

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### B) Coordinates of point \( Q \):

Point \( Q \) is symmetric to \( N \) with respect to the origin. The coordinates of \( Q \) are the negation of the coordinates of \( N \). Hence, if \( N(1, -2) \), then:
- \( Q(-1, 2) \)

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### C) Area of triangle \( MNP \):

We can calculate the area of triangle \( \Delta MNP \) using the formula for the area of a triangle with given vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):

\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Substituting \( M(-4, 3) \), \( N(1, -2) \), and \( P(4, 2) \):

\[
\text{Area} = \frac{1}{2} \left| -4(-2 - 2) + 1(2 - 3) + 4(3 + 2) \right|
\]
\[
= \frac{1}{2} \left| -4(-4) + 1(-1) + 4(5) \right|
\]
\[
= \frac{1}{2} \left| 16 - 1 + 20 \right|
\]
\[
= \frac{1}{2} \left| 35 \right|
\]
\[
\text{Area} = \frac{35}{2} = 17.5 \, \text{square units}
\]

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### Summary of answers:
- A) \( M(-4, 3) \), \( N(1, -2) \), \( P(4, 2) \)
- B) \( Q(-1, 2) \)
- C) Area of triangle \( MNP \) is \( 17.5 \) square units

Would you like more details or have further questions?

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### Follow-up questions:
1. What is the formula for finding the area of a triangle given its vertices in the coordinate plane?
2. Can you explain why the coordinates of \( Q \) are the negation of \( N \)'s coordinates?
3. How would the area change if \( N \) was at a different point?
4. Can you compute the perimeter of triangle \( MNP \)?
5. How does symmetry help when reflecting points across the origin?

**Tip:** To reflect a point across the origin, just negate both the x- and y-coordinates of the point.

In a coordinate system Oxy, the points M, N, and P are given, and we are asked to find the coordinates of these points, the symmetric point of N relative to the origin, and the area of triangle MNP.

Solve a coordinate geometry problem involving points M, N, and P. Find their coordinates, reflect point N across the origin, and compute the area of triangle MNP using key geometry formulas. This problem helps students understand symmetry and area calculations, suitable for Grades 7-9.

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