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Geometry and Trigonometry Difficulty: Medium

The figure presents two right triangles, A, B C and D E F. In triangle A, B C, side A, C is horizontal and side B C is drawn such that it is perpendicular to side A, C and point B, is above point C. Angle A, is labeled 32 degrees and the right angle symbol is indicated at angle C. In triangle D E F, side D F is horizontal, side E F is drawn such that it is perpendicular to side D F, and point E, is above point F. Angle D is labeled 58 degrees and the right angle symbol is indicated at angle F.

Triangles ABC and DEF are shown above. Which of the following is equal to the ratio $B C over A, B$ ?

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Explanation

Choice B is correct. In right triangle ABC, the measure of angle B must be 58° because the sum of the measure of angle A, which is 32°, and the measure of angle B is 90°. Angle D in the right triangle DEF has measure 58°. Hence, triangles ABC and DEF are similar (by angle-angle similarity). Since $side B C$ is the side opposite to the angle with measure 32° and AB is the hypotenuse in right triangle ABC, the ratio $the length of side B C over the length of side A, B$ is equal to $the length of side D F over the length of side D E$ .

Alternate approach: The trigonometric ratios can be used to answer this question. In right triangle ABC, the ratio $the length of side B C over the length of side A, B equals, the sine of 32 degrees$ . The angle E in triangle DEF has measure 32° because $the measure of angle D, plus the measure of angle E, equals 90 degrees$ . In triangle DEF, the ratio $the length of side D F over the length of side D E, equals the sine of 32 degrees$ . Therefore, $the length of side D F over the length of side D E, equals, the length of side B C over the length of side A, B$ .

Choice A is incorrect because $the length of side D E over the length of side D F$ is the reciprocal of the ratio $the length of side B C over the length of side A, B$ . Choice C is incorrect because $the length of side D F over the length of side D E, equals the length of side B C over the length of side A, C$ , not $the length of side B C over the length of side A, B$ . Choice D is incorrect because $the length of side E F over the length of side D E, equals, the length of side A, C over the length of side A, B$ , not $the length of side B C over the length of side A, B$ .