Math Problem Statement

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.

Solution

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

Given:

In a coordinate system OxyOxy, with a unit scale where 1 cm represents 1 unit, the points MM, NN, and PP are provided.

Questions:

  1. A) Write down the coordinates of points MM, NN, and PP.
  2. B) Write down the coordinates of point QQ, which is symmetric to point NN with respect to the origin.
  3. C) Find the area of triangle ΔMNP\Delta MNP.

Let's solve these step by step:


A) Coordinates of points MM, NN, and PP:

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

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

B) Coordinates of point QQ:

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

  • Q(1,2)Q(-1, 2)

C) Area of triangle MNPMNP:

We can calculate the area of triangle ΔMNP\Delta MNP using the formula for the area of a triangle with given vertices (x1,y1)(x_1, y_1), (x2,y2)(x_2, y_2), and (x3,y3)(x_3, y_3):

Area=12x1(y2y3)+x2(y3y1)+x3(y1y2)\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)M(-4, 3), N(1,2)N(1, -2), and P(4,2)P(4, 2):

Area=124(22)+1(23)+4(3+2)\text{Area} = \frac{1}{2} \left| -4(-2 - 2) + 1(2 - 3) + 4(3 + 2) \right| =124(4)+1(1)+4(5)= \frac{1}{2} \left| -4(-4) + 1(-1) + 4(5) \right| =12161+20= \frac{1}{2} \left| 16 - 1 + 20 \right| =1235= \frac{1}{2} \left| 35 \right| Area=352=17.5square units\text{Area} = \frac{35}{2} = 17.5 \, \text{square units}


Summary of answers:

  • A) M(4,3)M(-4, 3), N(1,2)N(1, -2), P(4,2)P(4, 2)
  • B) Q(1,2)Q(-1, 2)
  • C) Area of triangle MNPMNP is 17.517.5 square units

Would you like more details or have further questions?


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 QQ are the negation of NN's coordinates?
  3. How would the area change if NN was at a different point?
  4. Can you compute the perimeter of triangle MNPMNP?
  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.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Symmetry
Area of a Triangle

Formulas

Symmetry across the origin: (x, y) -> (-x, -y)
Area of triangle given vertices: A = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Theorems

Symmetry Theorem
Triangle Area Formula

Suitable Grade Level

Grades 7-9