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Difficulty: HardArea of Two-Dimensional Shapes

In the xyxy-plane, a square has vertices at (0,0)(0,0), (10,0)(10,0), (10,10)(10,10), and (0,10)(0,10). A line that passes through the points (0,2)(0,2) and (8,10)(8,10) divides the square into two regions. What is the area of the larger region?

Answer: 68

Answer

The area of the larger region is 68.
The total area of the square is 102=10010^2 = 100. The line segment between (0,2)(0,2) on the left boundary and (8,10)(8,10) on the top boundary forms a right triangle with the top-left vertex of the square (0,10)(0,10). The legs of this right triangle have lengths 10−2=810 - 2 = 8 and 8−0=88 - 0 = 8, so its area is 12×8×8=32\frac{1}{2} \times 8 \times 8 = 32. The area of the other region is 100−32=68100 - 32 = 68. The larger area is therefore 68.

Step-by-Step Solution

1
Calculate the area of the square
100
The square has vertices at (0,0)(0,0), (10,0)(10,0), (10,10)(10,10), and (0,10)(0,10), which gives a side length of 10. The area of a square is side2\text{side}^2.
2
Determine the dimensions of the smaller triangular region formed by the line
Legs of length 8 and 8
The line intersects the left edge at (0,2)(0,2) and the top edge at (8,10)(8,10). The corner of the square is at (0,10)(0,10). The distance from (0,2)(0,2) to (0,10)(0,10) is 8, and the distance from (8,10)(8,10) to (0,10)(0,10) is 8.
3
Calculate the area of the smaller triangular region
32
The area of a right triangle is 12×base×height=12×8×8=32\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 8 = 32.
4
Subtract the area of the smaller region from the total area of the square to find the area of the larger region
68
The area of the larger region is the total area of the square minus the area of the smaller region: 100−32=68100 - 32 = 68.

Key Concept

Calculating the area of a region by partitioning a geometric shape or using subtraction of areas.
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Topics

Area of Two-Dimensional Shapes

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