Introduction to Rectangle Properties
When working with geometric figures like rectangles, understanding the properties of these shapes is essential. Specifically, in a rectangle, opposite sides are equal in length, and the diagonals bisect each other. This guide will walk you through the process of finding the coordinates of point D in a rectangle ABCD, given points A(7, 4), B(10, 2), C(6, -4).
Step-by-Step Guide to Finding the Coordinates of Point D
To find the coordinates of point D in rectangle ABCD, we will start by using the properties of the rectangle and the midpoint formula.
Step 1: Calculate the Midpoint of Diagonal AC
The first step involves finding the midpoint of diagonal AC. The formula for the midpoint of two points (x1, y1) and (x2, y2) is: [text{Midpoint} left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right)]
Substituting the coordinates of A(7, 4) and C(6, -4), we get:
[text{Midpoint} left(frac{7 6}{2}, frac{4 (-4)}{2}right) left(frac{13}{2}, 0right) 6.5, 0]Step 2: Use the Midpoint to Find Point D
The midpoint of diagonal BD is the same as the midpoint of diagonal AC. Therefore, we can set up the equation for the midpoint of BD as follows:
[left(frac{10 x_D}{2}, frac{2 y_D}{2}right) (6.5, 0)]This gives us two separate equations to find x_D and y_D:
For the x-coordinates:
[frac{10 x_D}{2} 6.5]Multiplying both sides by 2:
[10 x_D 13]Subtracting 10 from both sides:
[x_D 3]For the y-coordinates:
[frac{2 y_D}{2} 0]Multiplying both sides by 2:
[2 y_D 0]Subtracting 2 from both sides:
[y_D -2]Therefore, the coordinates of point D are:
[boxed{(3, -2)}]Alternative Methods and Verification
Alternatively, you can also use the slopes and vector approach:
Slope of BC is calculated as:
[text{Slope} frac{2 - (-4)}{10 - 6} frac{6}{4} frac{3}{2}]Slope of AD is the same as BC because AD is parallel to BC. Hence:
[text{Slope of } AD frac{y_D - 4}{x_D - 7} frac{3}{2}]Since A(7, 4) must move -3 in x and -2 in y to align with point C(6, -4) and to find point D:
[text{D coordinates: } (7 - 3, 4 - 2) (3, -2)]Thus, the coordinates of point D are indeed (3, -2).
Conclusion
Understanding the properties of rectangles and using the midpoint formula or vector methods are powerful tools to find unknown coordinates in such geometric shapes. This approach not only solves the problem but also reinforces the underlying geometric principles.