Question
A rectangle has its vertices at the following coordinates: \(A(1, 2)\), \(B(1, 5)\), \(C(4, 5)\), and \(D(4, 2)\). What are the coordinates of the center of the rectangle?
Ask by Gray Zimmerman. in South Africa
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The center of the rectangle is at (2.5, 3.5).
Solution
To find the center of the rectangle with vertices \(A(1, 2)\), \(B(1, 5)\), \(C(4, 5)\), and \(D(4, 2)\), we can use the midpoint formula for one of the diagonals. The diagonals of a rectangle bisect each other, so the center will be the midpoint of either diagonal \(AC\) or \(BD\).
Let's calculate the midpoint of diagonal \(AC\):
1. **Coordinates of \(A\)**: \((1, 2)\)
2. **Coordinates of \(C\)**: \((4, 5)\)
The midpoint \(M\) is given by:
\[
M = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{1 + 4}{2}, \frac{2 + 5}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right)
\]
So, the coordinates of the center of the rectangle are \(\left( \frac{5}{2}, \frac{7}{2} \right)\), which can also be written as \((2.5, 3.5)\).
**Answer:** The center of the rectangle is ( 2.5, 3.5 ).
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Simplify this solution Beyond the Answer
To find the coordinates of the center of the rectangle, you can average the x-coordinates and the y-coordinates of the vertices. The vertices are \(A(1, 2)\), \(B(1, 5)\), \(C(4, 5)\), and \(D(4, 2)\). 1. Average of x-coordinates: \(\frac{1 + 1 + 4 + 4}{4} = \frac{10}{4} = 2.5\) 2. Average of y-coordinates: \(\frac{2 + 5 + 5 + 2}{4} = \frac{14}{4} = 3.5\) Thus, the coordinates of the center of the rectangle are \((2.5, 3.5)\).
