Math Problem Statement

The midpoint M of CD has coordinates (0.5, –3.5). Point D has coordinates (–13, –18). Find the coordinates of point C.

Write the coordinates as decimals or integers.

Solution

To find the coordinates of point C, we can use the midpoint formula. The midpoint MM of a line segment with endpoints C(x1,y1)C(x_1, y_1) and D(x2,y2)D(x_2, y_2) is given by the formula:

M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

We are given the midpoint M=(0.5,3.5)M = (0.5, -3.5) and the coordinates of point D D=(13,18)D = (-13, -18). Let the coordinates of point C be (x1,y1)(x_1, y_1). We can set up the equations for the midpoint:

x1+(13)2=0.5(for the x-coordinate)\frac{x_1 + (-13)}{2} = 0.5 \quad \text{(for the x-coordinate)} y1+(18)2=3.5(for the y-coordinate)\frac{y_1 + (-18)}{2} = -3.5 \quad \text{(for the y-coordinate)}

Solving for the x-coordinate:

x1132=0.5\frac{x_1 - 13}{2} = 0.5 Multiply both sides by 2: x113=1x_1 - 13 = 1 Add 13 to both sides: x1=14x_1 = 14

Solving for the y-coordinate:

y1182=3.5\frac{y_1 - 18}{2} = -3.5 Multiply both sides by 2: y118=7y_1 - 18 = -7 Add 18 to both sides: y1=11y_1 = 11

Thus, the coordinates of point C are (14,11)(14, 11).

Let me know if you'd like further clarification or if you have any questions!


Here are 5 related questions to expand on this:

  1. How would the midpoint formula change if the midpoint had been given as a point in three-dimensional space?
  2. What would happen to the coordinates of point C if point D had different coordinates, say (1,4)(1, 4)?
  3. Can the midpoint of a line segment be used to find coordinates in a triangle, and if so, how?
  4. How can you use the midpoint formula to check if a set of points is symmetrically distributed?
  5. How would you use the distance formula to find the length of the line segment between points C and D?

Tip: Always check your calculations by substituting the values back into the midpoint formula to verify the result!

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Math Problem Analysis

Mathematical Concepts

Midpoint Formula
Coordinate Geometry
Algebra

Formulas

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

Theorems

Midpoint Theorem

Suitable Grade Level

Grades 7-9