Math Problem Statement

The midpoint M of PQ has coordinates (3.5, –4.5). Point Q has coordinates (–2, –6). Find the coordinates of point P.

Write the coordinates as decimals or integers.

Solution

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

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

Given:

  • Midpoint M(3.5,4.5)M(3.5, -4.5)
  • Point Q(2,6)Q(-2, -6)

We can set up the following equations based on the midpoint formula:

  1. x1+(2)2=3.5\frac{x_1 + (-2)}{2} = 3.5
  2. y1+(6)2=4.5\frac{y_1 + (-6)}{2} = -4.5

Step 1: Solve for x1x_1

Multiply both sides of the first equation by 2:

x12=7x_1 - 2 = 7

Add 2 to both sides:

x1=9x_1 = 9

Step 2: Solve for y1y_1

Multiply both sides of the second equation by 2:

y16=9y_1 - 6 = -9

Add 6 to both sides:

y1=3y_1 = -3

Final Answer:

The coordinates of point PP are (9,3)(9, -3).

Would you like more details or have any questions about the steps?


Here are five related questions:

  1. How would the midpoint change if point PP was moved further away from QQ?
  2. What is the distance between points PP and QQ?
  3. Can you find the slope of the line passing through points PP and QQ?
  4. How would you find the coordinates of a point that divides the segment PQPQ in a different ratio?
  5. What are the coordinates of the midpoint if QQ were at (5,2)(-5, 2)?

Tip: Always verify your answers by substituting the coordinates back into the midpoint formula to check for consistency.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Midpoint Formula

Formulas

Midpoint formula

Theorems

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Suitable Grade Level

Grades 7-9