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Difficulty: EasyGeometric Figures on the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, a rectangle has vertices at (1,2)(1, 2), (7,2)(7, 2), (7,10)(7, 10), and (1,10)(1, 10). What is the length of a diagonal of this rectangle?

Answer: 10

Answer

The length of a diagonal of the rectangle is 10.
The width of the rectangle is 7−1=67 - 1 = 6, and the height is 10−2=810 - 2 = 8. The diagonal forms the hypotenuse of a right triangle with legs of 6 and 8. By the Pythagorean theorem, the length of the diagonal is 62+82=36+64=100=10\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.

Step-by-Step Solution

1
Calculate the horizontal width of the rectangle.
Width = 7−1=67 - 1 = 6
The horizontal distance is found by subtracting the x-coordinates of the horizontal vertices.
2
Calculate the vertical height of the rectangle.
Height = 10−2=810 - 2 = 8
The vertical distance is found by subtracting the y-coordinates of the vertical vertices.
3
Apply the Pythagorean theorem to find the diagonal length.
Diagonal length = 62+82=10\sqrt{6^2 + 8^2} = 10
The diagonal of a rectangle forms the hypotenuse of a right triangle with legs equal to the width and the height.

Key Concept

Using coordinate differences to find dimensions of a figure and using the Pythagorean theorem to find its diagonal length.
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Topics

Geometric Figures on the Coordinate Plane

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