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Difficulty: MediumGeometric Figures on the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, a square has two opposite vertices at (1,−3)(1, -3) and (7,5)(7, 5). What is the area of the square, in square units?

Answer: 50 square units

Answer

The area of the square is 50 square units.
The length of the diagonal of the square is found using the distance formula between the two opposite vertices: d=(7−1)2+(5−(−3))2=62+82=10d = \sqrt{(7 - 1)^2 + (5 - (-3))^2} = \sqrt{6^2 + 8^2} = 10. The area of a square can be calculated using its diagonal length dd with the formula Area=d22\text{Area} = \frac{d^2}{2}. Substituting d=10d = 10 gives Area=1022=50\text{Area} = \frac{10^2}{2} = 50 square units.

Step-by-Step Solution

1
Calculate the length of the diagonal of the square using the distance formula.
The diagonal length is 1010.
The distance between opposite vertices of a square represents the length of its diagonal.
2
Determine the area of the square using the diagonal length.
The area is 5050.
The area of a square with diagonal dd is given by d22\frac{d^2}{2}.

Key Concept

Finding the area of a square on the coordinate plane using its diagonal.
Estimated Time:1m 30s
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Topics

Geometric Figures on the Coordinate Plane

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