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Geometry and Trigonometry Difficulty: Hard

A graphic designer is creating a logo for a company. The logo is shown in the figure above. The logo is in the shape of a trapezoid and consists of three congruent equilateral triangles. If the perimeter of the logo is 20 centimeters, what is the combined area of the shaded regions, in square centimeters, of the logo?

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Explanation

Choice C is correct. It’s given that the logo is in the shape of a trapezoid that consists of three congruent equilateral triangles, and that the perimeter of the trapezoid is 20 centimeters (cm). Since the perimeter of the trapezoid is the sum of the lengths of 5 of the sides of the triangles, the length of each side of an equilateral triangle is $the fraction 20 over 5 equals 4 centimeters$ . Dividing up one equilateral triangle into two right triangles yields a pair of congruent 30°-60°-90° triangles. The shorter leg of each right triangle is half the length of the side of an equilateral triangle, or 2 cm. Using the Pythagorean Theorem, $a, squared, plus b squared, equals c squared$ , the height of the equilateral triangle can be found. Substituting $a, equals 2$ and $c equals 4$ and solving for b yields $the square root of, 4 squared, minus 2 squared, end root, equals the square root of 12, which equals, 2 times the square root of 3 centimeters$ cm. The area of one equilateral triangle is $one half b h$ , where $b equals 2$ and $h equals, 2 times the square root of 3$ . Therefore, the area of one equilateral triangle is $one half times 4, times, open parenthesis, 2 times the square root of 3, close parenthesis, equals, 4 times the square root of 3 centimeters squared$ . The shaded area consists of two such triangles, so its area is $2 times 4, times the square root of 3, equals, 8 times the square root of 3 centimeters squared$ .

Alternate approach: The area of a trapezoid can be found by evaluating the expression $one half times, open parenthesis, b sub 1 plus b sub 2, close parenthesis, times h$ , where $b sub 1$ is the length of one base, $b sub 2$ is the length of the other base, and h is the height of the trapezoid. Substituting $b sub 1 equals 8$ , $b sub 2 equals 4$ , and $h equals, 2 times the square root of 3$ yields the expression $one half times, open parenthesis, 8 plus 4, close parenthesis, times, open parenthesis, 2 times the square root of 3, close parenthesis$ , or $one half times 12, times, open parenthesis, 2 times the square root of 3, close parenthesis$ , which gives an area of $12 times the square root of 3 centimeters squared$ for the trapezoid. Since two-thirds of the trapezoid is shaded, the area of the shaded region is $two thirds times, 12 times the square root of 3, equals, 8 times the square root of 3$ .

Choice A is incorrect. This is the height of the trapezoid. Choice B is incorrect. This is the area of one of the equilateral triangles, not two. Choice D is incorrect and may result from using a height of 4 for each triangle rather than the height of $2 times the square root of 3$ .