What is the Fourth Vertex of a Rectangle?

When working with geometric shapes like rectangles, it is often useful to find the coordinates of the fourth vertex when three vertices are given. This article will walk you through the process using a detailed example and explain the relevant mathematical properties used in the calculations.

Understanding the Properties of a Rectangle

A rectangle is a quadrilateral with four right angles. It possesses several geometric properties that we can utilize to find the coordinates of the fourth vertex. Notably:

Opposite sides are equal in length. Diagonals bisect each other. Adjacent sides are perpendicular to each other.

Step-by-Step Guide to Finding the Fourth Vertex

Let's take an example where three vertices of a rectangle are given as A(3, 4), B(-1, 2), and C(2, -4). We need to find the fourth vertex D(x, y).

Step 1: Calculate the Midpoint of the Diagonal AC

The midpoint of a diagonal is the point that divides the diagonal into two equal parts. The midpoint formula is:

MAC (left( frac{x_A x_C}{2}, frac{y_A y_C}{2} right))

Using the coordinates of A and C:

MAC (left( frac{3 2}{2}, frac{4 (-4)}{2} right) left( frac{5}{2}, 0 right))

Step 2: Set the Midpoint of BD Equal to AC

The midpoint of the other diagonal BD must be the same as the midpoint of AC, as this property holds true for a rectangle.

The midpoint of BD is:

MBD (left( frac{x_B x_D}{2}, frac{y_B y_D}{2} right))

Since the midpoints are equal:

(left( frac{-1 x}{2}, frac{2 y}{2} right) left( frac{5}{2}, 0 right))

Step 3: Solve for x and y

To find the coordinates of D, we solve:

(frac{-1 x}{2} frac{5}{2})

Multiplying both sides by 2:

-1 x 5

Solving for x:

x 6

For the y-coordinate:

(frac{2 y}{2} 0)

Multiplying both sides by 2:

2 y 0

Solving for y:

y -2

Conclusion

The coordinates of the fourth vertex D are (6, -2).

Additional Verification Using Slope and Length

For further confirmation, the slopes and lengths of the sides can be checked.

Checking the Slopes for Perpendicularity

slope AB (frac{2 - 4}{3 - (-1)} frac{-2}{4} -frac{1}{2})

slope BC (frac{1 - 2}{4 - 3} frac{-1}{1} -1)

Since the product of these slopes is -1:

((- frac{1}{2}) times (-1) 1)

They are mutually perpendicular as expected for adjacent sides of a rectangle.

Verifying Side Lengths

The length of AB:

AB (sqrt{(3 - (-1))^2 (4 - 2)^2} sqrt{4^2 2^2} sqrt{16 4} sqrt{20} 2sqrt{5})

The length of BC:

BC (sqrt{(4 - 3)^2 (1 - 2)^2} sqrt{1^2 (-1)^2} sqrt{1 1} sqrt{2})

The length of CD (using D(6, -2)):

CD (sqrt{(6 - 2)^2 (-2 - 1)^2} sqrt{4^2 (-3)^2} sqrt{16 9} sqrt{25} 5)

The length of DA:

DA (sqrt{(6 - (-1))^2 (-2 - 2)^2} sqrt{7^2 (-4)^2} sqrt{49 16} sqrt{65})

Thus, the properties of the sides confirm that the calculated coordinates for D are correct.

Conclusion

The fourth vertex of the rectangle, given three vertices A(3, 4), B(-1, 2), and C(2, -4), is D(6, -2). The geometric properties and calculations using the midpoint and slopes verified the solution.