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Geometry and Trigonometry / Lines, angles, and triangles Difficulty: Hard

The figure presents right triangle M N P. Side M P is horizontal, with vertex M to the left of vertex P. Vertex N lies above side M P, and angle N is a right angle. Point Q lies on horizontal side M P directly below vertex N. Vertical line segment N Q is drawn, forming right triangles M N Q and Q N P. Side M N is labeled 3 and side N P is labeled 4.

In the figure above, what is the length of $line segment NQ$ ?

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Explanation

Choice C is correct. First, $line segment M P$ is the hypotenuse of right $triangle M N P$ , whose legs have lengths 3 and 4. Therefore, $open parenthesis, the length of line segment M P, close parenthesis, squared, equals, 3 squared plus 4 squared$ , so $open parenthesis, the length of line segment M P, close parenthesis, squared, equals 25$ and $the length of line segment M P equals 5$ . Second, because $angle M N P$ corresponds to $angle N Q P$ and because $angle M P N$ corresponds to $angle N P Q$ , $triangle M N P$ is similar to $triangle N Q P$ . The ratio of corresponding sides of similar triangles is constant, so $the length of line segment N Q over the length of line segment M N equals the length of line segment N P over the length of line segment M P$ . Since $M P equals 5$ and it’s given that $the length of line segment M N equals 3$ and $the length of line segment N P equals 4$ , $the length of line segment N Q, over 3, equals 4 over 5$ . Solving for NQ results in $the length of line segment N Q equals 12 over 5$ , or 2.4.

Choices A, B, and D are incorrect and may result from setting up incorrect ratios.