Math Problem Statement

ABCD is a parallelogram. What are the coordinates of point D? Given points are A(-11, 23), B(-13, 29), and C(5, 22).

Solution

The image shows a parallelogram ABCD on a coordinate plane with the points labeled as follows:

  • Point A (-11, 23)
  • Point B (-13, 29)
  • Point C (5, 22)

You are asked to find the coordinates of point D.

Solution:

In a parallelogram, opposite sides are parallel and equal in length. Thus, the vector from A to B is equal to the vector from D to C. We can find the coordinates of point D by using this property.

The vector from A to B is: AB=BA=(13(11),2923)=(2,6)\overrightarrow{AB} = B - A = (-13 - (-11), 29 - 23) = (-2, 6)

Since AB=DC\overrightarrow{AB} = \overrightarrow{DC}, the vector from D to C must also be (-2, 6). Therefore, to find D, we can apply the vector to point C: D=CAB=(5,22)(2,6)=(5+2,226)=(7,16)D = C - \overrightarrow{AB} = (5, 22) - (-2, 6) = (5 + 2, 22 - 6) = (7, 16)

Thus, the coordinates of point D are (7, 16).

Let me know if you need further clarification or details! Also, here are a few related questions:

  1. What is the general formula for finding the coordinates of a point in a parallelogram?
  2. How do you find the area of a parallelogram given its vertex coordinates?
  3. What is the distance between two points in a coordinate plane?
  4. How do you determine if a quadrilateral is a parallelogram from its vertex coordinates?
  5. How can you prove that opposite sides of a parallelogram are parallel using vectors?

Tip: When dealing with parallelograms, always remember that opposite sides are not only parallel but also equal in length. This can help simplify many problems like this one.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Vectors
Properties of Parallelograms

Formulas

Vector addition/subtraction formula: (x2 - x1, y2 - y1)

Theorems

Opposite sides of a parallelogram are equal and parallel

Suitable Grade Level

Grade 9-10