Math Problem Statement

If (6, −8) is the midpoint of segment PQ and the coordinates of Q are (−5, −12), find the coordinates of P

Solution

We are given that the midpoint of segment PQPQ is (6,8)(6, -8), and the coordinates of point QQ are (5,12)(-5, -12). We need to find the coordinates of point P(x1,y1)P(x_1, y_1).

The formula for the midpoint of a segment is: (x1+x22,y1+y22)\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) where (x1,y1)(x_1, y_1) are the coordinates of point PP, and (x2,y2)(x_2, y_2) are the coordinates of point QQ.

Using the given midpoint (6,8)(6, -8) and Q(5,12)Q(-5, -12), we can set up the following equations:

x1+(5)2=6\frac{x_1 + (-5)}{2} = 6 y1+(12)2=8\frac{y_1 + (-12)}{2} = -8

Solving for x1x_1:

x152=6\frac{x_1 - 5}{2} = 6 Multiply both sides by 2: x15=12x_1 - 5 = 12 Add 5 to both sides: x1=17x_1 = 17

Solving for y1y_1:

y1122=8\frac{y_1 - 12}{2} = -8 Multiply both sides by 2: y112=16y_1 - 12 = -16 Add 12 to both sides: y1=4y_1 = -4

Thus, the coordinates of point PP are (17,4)(17, -4).

Do you need further clarification or details? Here are some related questions for practice:

  1. What is the formula for the midpoint of a segment?
  2. How do you find the distance between two points?
  3. If Q(3,5)Q(3, 5) and P(7,9)P(7, 9), what is the midpoint of segment PQPQ?
  4. What is the slope of the line passing through points P(17,4)P(17, -4) and Q(5,12)Q(-5, -12)?
  5. Can a line segment have a negative midpoint?

Tip: Always remember, the midpoint is simply the average of the x-coordinates and y-coordinates of the two points.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Midpoint Formula
Algebra

Formulas

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

Theorems

Midpoint Theorem

Suitable Grade Level

Grades 8-10