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Difficulty: MediumGeometric Figures on the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, three vertices of a rectangle are A(−4,1)A(-4, 1), B(2,9)B(2, 9), and C(6,6)C(6, 6). What is the area of the rectangle, in square units?

Answer: 50

Answer

50
The length of side ABAB is calculated as (2−(−4))2+(9−1)2=36+64=10\sqrt{(2 - (-4))^2 + (9 - 1)^2} = \sqrt{36 + 64} = 10. The length of the adjacent side BCBC is calculated as (6−2)2+(6−9)2=16+9=5\sqrt{(6 - 2)^2 + (6 - 9)^2} = \sqrt{16 + 9} = 5. The area of the rectangle is the product of these two perpendicular side lengths, which is 10×5=5010 \times 5 = 50.

Step-by-Step Solution

1
Calculate the length of side ABAB using the distance formula.
AB=10AB = 10
To find one of the side lengths of the rectangle.
2
Calculate the length of side BCBC using the distance formula.
BC=5BC = 5
To find the adjacent side length of the rectangle.
3
Multiply the two adjacent side lengths to find the area of the rectangle.
50
The area of a rectangle is equal to the product of its length and width.

Key Concept

Calculating the area of a geometric figure on the coordinate plane by determining its side lengths using the distance formula.
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Topics

Geometric Figures on the Coordinate Plane

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