Introduction

In this article, we will explore how to find the coordinates of vertex D in a parallelogram ABCD. We will use the properties of a parallelogram and the concept of midpoints to solve the problem. By understanding these geometric principles, you will be able to tackle similar problems with confidence.

Problem Statement

Given the vertices of parallelogram ABCD are A(1, 4), B(2, 7), and C(-1, 3), we are to find the coordinates of vertex D.

Using the Midpoint Property

A key property of parallelograms is that their diagonals bisect each other. This means that the midpoint of diagonal AC is the same as the midpoint of diagonal BD. Let's use this property to find the coordinates of vertex D.

Step 1: Calculate the Midpoint of AC

The coordinates of points A and C are A(1, 4) and C(-1, 3).

The midpoint MAC is calculated as follows:

M_{AC} left(frac{x_A x_C}{2}, frac{y_A y_C}{2}right) left(frac{1 (-1)}{2}, frac{4 3}{2}right) left(frac{0}{2}, frac{7}{2}right) (0, 3.5)

Step 2: Set Up the Midpoint of BD

The coordinates of point B are B(2, 7). Let the coordinates of point D be D(x, y).

The midpoint MBD is given by:

M_{BD} left(frac{x_B x_D}{2}, frac{y_B y_D}{2}right) left(frac{2 x}{2}, frac{7 y}{2}right)

Step 3: Set the Midpoints Equal

Since MAC MBD, we have:

left(frac{2 x}{2}, frac{7 y}{2}right) (0, 3.5)

This gives us two equations:

frac{2 x}{2} 0 quad text{and} quad frac{7 y}{2} 3.5

Step 4: Solve the Equations

From the first equation:

2 x 0 quad Rightarrow quad x -2

From the second equation:

7 y 7 quad Rightarrow quad y 0

Therefore, the coordinates of vertex D are D(-2, 0).

Slope Method

Alternatively, we can use the slope method to find the coordinates of vertex D.

Step 1: Calculate Slopes

The slopes of the sides AB, BC, and CD can be calculated as follows:

Slope AB frac{7 - 4}{2 - 1} 3

Slope BC frac{3 - 7}{-1 - 2} frac{-4}{-3} frac{4}{3}

Using the slopes, we can find the equations of the lines:

y - 4 3(x - 1) quad Rightarrow quad 3x - y -1

For line CD (parallel to AB):

y - 3 3(x 1) quad Rightarrow quad 3x - y -6

For line BC (parallel to AD):

y - 7 frac{4}{3}(x - 2) quad Rightarrow quad 3y - 21 4x - 8 quad Rightarrow quad 4x - 3y 13

For line AD (parallel to BC):

y - 4 frac{4}{3}(x - 1) quad Rightarrow quad 4x - 3y 8

Step 2: Find the Intersection

The coordinates of D can be found by solving the equations of CD and AD:

3x - y -6

4x - 3y 8

Solving these equations, we find:

x -2 quad text{and} quad y 0

Therefore, the coordinates of vertex D are D(-2, 0).

Conclusion

By using the midpoint property and the slope method, we have determined that the coordinates of vertex D in parallelogram ABCD are D(-2, 0).