Finding the Coordinates of Points in Triangle PQR Using Midpoints

Given the midpoints of the sides of a triangle, we can use the midpoint formula to find the coordinates of the vertices. In this article, we will solve a specific problem where the coordinates of the midpoints of sides P Q, Q R, and P R in triangle PQR are (-2, 3), (5, -1), and (-4, -7) respectively. We will walk through the process step by step.

Understanding the Midpoint Formula

The midpoint formula states that the midpoint M of a line segment connecting points (x1, y1) and (x2, y2) is given by:

M (x1 x2)/2, (y1 y2)/2

Applying the Midpoint Formula to the Given Problem

We are given the midpoints of the sides of triangle PQR:

A (-2, 3), where A is the midpoint of PQ B (5, -1), where B is the midpoint of QR C (-4, -7), where C is the midpoint of PR

Let the coordinates of points P, Q, and R be (x1, y1), (x2, y2), and (x3, y3) respectively.

Midpoint Equations

Using the midpoint formula, we can create the following equations:

A ((x1 x2)/2, (y1 y2)/2) (-2, 3) B ((x2 x3)/2, (y2 y3)/2) (5, -1) C ((x1 x3)/2, (y1 y3)/2) (-4, -7)

These give us the following equations:

(x1 x2)/2 -2 rarr; x1 x2 -4 (y1 y2)/2 3 rarr; y1 y2 6 (x2 x3)/2 5 rarr; x2 x3 10 (y2 y3)/2 -1 rarr; y2 y3 -2 (x1 x3)/2 -4 rarr; x1 x3 -8 (y1 y3)/2 -7 rarr; y1 y3 -14

Solving the System of Equations

We now have a system of six equations to solve for the coordinates of points P, Q, and R. We will solve these step by step:

For x-coordinates:

x1 x2 -4 x2 x3 10 x1 x3 -8

For y-coordinates:

y1 y2 6 y2 y3 -2 y1 y3 -14

Solving for x-coordinates:

From the first equation, we can express x2 in terms of x1:

x2 -4 - x1

Substitute this into the second equation:

-4 - x1 x3 10 rarr; x3 14 - x1

Substitute this into the third equation:

x1 (14 - x1) -8 rarr; 14 -8

This simplifies to:

2x1 -22 rarr; x1 -11

Using x1 -11 in the first equation:

x2 -4 - (-11) 7

Using x1 -11 in the third equation:

x3 14 - (-11) 3

Solving for y-coordinates:

From the first equation, we can express y2 in terms of y1:

y2 6 - y1

Substitute this into the second equation:

6 - y1 y3 -2 rarr; y3 -8 - y1

Substitute this into the third equation:

y1 (-8 - y1) -14 rarr; -8 -14

This simplifies to:

2y1 -6 rarr; y1 -3

Using y1 -3 in the first equation:

y2 6 - (-3) 9

Using y1 -3 in the second equation:

y3 -8 - (-3) -11

Final Answer

The coordinates of points P, Q, and R are:

P (-11, -3) Q (7, 9) R (3, -11)

Therefore, the final coordinates are:

P (-11, -3) Q (7, 9) R (3, -11)

This concludes our detailed solution to the problem of finding the coordinates of points in triangle PQR using midpoints.