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Geometry and Trigonometry / Area and volume Difficulty: Medium

The figure presents the graph of square A B C D in the x y plane. The numbers negative 5 and 5 are indicated on each axis. The coordinates of each vertex are as follows.
Vertex A has coordinates 0 comma negative 5.
Vertex B has coordinates negative 5 comma 0.
Vertex C has coordinates 0 comma 5.
Vertex D has coordinates 5 comma 0.

In the xy-plane shown, square ABCD has its diagonals on the x- and y-axes. What is the area, in square units, of the square?

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Explanation

Choice C is correct. The two diagonals of square ABCD divide the square into 4 congruent right triangles, where each triangle has a vertex at the origin of the graph shown. The formula for the area of a triangle is $A equals, one half times b h$ , where b is the base length of the triangle and h is the height of the triangle. Each of the 4 congruent right triangles has a height of 5 units and a base length of 5 units. Therefore, the area of each triangle is $A equals, one half times, 5 times 5$ , or 12.5 square units. Since the 4 right triangles are congruent, the area of each is $one fourth$ of the area of square ABCD. It follows that the area of the square ABCD is equal to $4 times 12 point 5$ , or 50 square units.

Choices A and D are incorrect and may result from using 5 or 25, respectively, as the area of one of the 4 congruent right triangles formed by diagonals of square ABCD. However, the area of these triangles is 12.5. Choice B is incorrect and may result from using 5 as the length of one side of square ABCD. However, the length of a side of square ABCD is $5 times the square root of 2$ .