Math Problem Statement

If ABCD is a parallelogram with vertices at A(-4,0), B(-3,3), and C(2,2), find the coordinates of point D.

Solution

The problem asks to find the coordinates of point DD in parallelogram ABCDABCD with the given vertices A(4,0)A(-4,0), B(3,3)B(-3,3), and C(2,2)C(2,2).

Approach:

In a parallelogram, opposite sides are parallel and equal in length, so the diagonals bisect each other. This property can help find point DD.

  1. Let's denote the midpoint of diagonal ACAC and diagonal BDBD.
  2. The midpoint of both diagonals must be the same.

Step 1: Calculate the midpoint of ACAC

  • The midpoint formula for two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
  • Applying this to points A(4,0)A(-4, 0) and C(2,2)C(2, 2): MAC=(4+22,0+22)=(1,1)M_{AC} = \left( \frac{-4 + 2}{2}, \frac{0 + 2}{2} \right) = (-1, 1)

Step 2: Set the midpoint of BDBD equal to MACM_{AC}

  • Let the coordinates of point DD be (xD,yD)(x_D, y_D).
  • The midpoint of BDBD is: MBD=(3+xD2,3+yD2)M_{BD} = \left( \frac{-3 + x_D}{2}, \frac{3 + y_D}{2} \right)
  • Since MAC=MBDM_{AC} = M_{BD}, equate the coordinates: 3+xD2=1and3+yD2=1\frac{-3 + x_D}{2} = -1 \quad \text{and} \quad \frac{3 + y_D}{2} = 1

Step 3: Solve the system of equations

  • From 3+xD2=1\frac{-3 + x_D}{2} = -1: 3+xD=2xD=1-3 + x_D = -2 \quad \Rightarrow \quad x_D = 1
  • From 3+yD2=1\frac{3 + y_D}{2} = 1: 3+yD=2yD=13 + y_D = 2 \quad \Rightarrow \quad y_D = -1

Final Answer:

The coordinates of point DD are (1,1)(1, -1).

Would you like any additional details or explanations on this?


Here are 5 related questions:

  1. How can we prove the diagonals of a parallelogram bisect each other?
  2. Can we use vector methods to solve this problem?
  3. How do you determine if four points form a parallelogram?
  4. What are the properties of diagonals in special parallelograms like rectangles or rhombuses?
  5. How do we find the area of a parallelogram given its vertices?

Tip: When solving geometry problems, sketching the figure often provides helpful visual insights.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Properties of Parallelograms

Formulas

Midpoint Formula M = ((x1 + x2)/2, (y1 + y2)/2)

Theorems

Diagonals of a parallelogram bisect each other

Suitable Grade Level

Grades 8-10