PSAT Math Multiple-Choice Question 137: Answer and Explanation

Question: 137

Note: Figure not drawn to scale.

In triangle LMN above, angle NPO is congruent to angle MLN, NP = 7, and LN = 18. If the length of MN is 1 unit less than 3 times the length of NO, what is the length of side NO?

• A. 11
• B. 5
• C.
• D.

Correct Answer: C

Explanation:

C

The question asks for the length of side on a figure. Use the geometry basic approach. Start by labeling the figure with the given information. Label NP as 7 and LN as 18. Mark angles NPO and MLN as congruent.

Since the two triangles have a congruent angle and share angle N, the third angles of the triangles will be congruent as well. Therefore, the triangles are similar, with triangle NPO corresponding to triangle NLM. The corresponding sides are difficult to see, since the figure is not drawn to scale and the angle in the lower left of NLM is congruent to the angle at the top of NPO. Redraw the two triangles separately using the description in the question to get the correct scale.

Similar triangles have proportional sides, so set up the proportion with the known sides compared to NO and its corresponding side MN: . Plug the information given for the lengths of NP and LN into the proportion to get . The question states that the length of is 1 unit less than 3 times the length of , so translate this into MN = 3(NO) - 1. Put the right side of this equation into the proportion for MN to get . To solve for NO, cross-multiply to get 18(NO) = 7[3(NO) - 1] or 18NO = 21NO - 7 Subtract 21NO from both sides to get -3NO = -7, then divide both sides by -3 to get . The correct answer is (C).