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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

The figure presents right triangle A, B C, with horizontal side A, C. Point B is directly above point A and angle A is a right angle. Point D lies on side A, B and point E lies on side B C. Horizontal line segment D E is drawn and angle D is a right angle.

In the figure above, $tangent of B equals the fraction 3 over 4$ . If $B C equals 15$ and $D A, equals 4$ , what is the length of $line segment D E$ ?

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Explanation

The correct answer is 6. Since $tangent of B equals three fourths$ , $triangle A, B C$ and $triangle D B E$ are both similar to 3-4-5 triangles. This means that they are both similar to the right triangle with sides of lengths 3, 4, and 5. Since $the length of side B C equals 15$ , which is 3 times as long as the hypotenuse of the 3-4-5 triangle, the similarity ratio of $triangle A, B C$ to the 3-4-5 triangle is 3:1. Therefore, the length of $side A, C$ (the side opposite to $angle B$ ) is $3 times 3, equals 9$ , and the length of $side A, B$ (the side adjacent to $angle B$ ) is $4 times 3, equals 12$ . It is also given that $the length of side D A, equals 4$ . Since $the length of side A, B equals, the length of side D A, plus the length of side D B$ and $the length of side A, B equals 12$ , it follows that $the length of side D B equals 8$ , which means that the similarity ratio of $triangle D B E$ to the 3-4-5 triangle is 2:1 ( $side D B$ is the side adjacent to $angle B$ ). Therefore, the length of $side D E$ , which is the side opposite to $angle B$ , is $3 times 2, equals 6$ .