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

In the figure, $\bar{L Q}$ intersects $\bar{M P}$ at point $R$ , and $\bar{L M}$ is parallel to $\bar{P Q}$ . The lengths of $\bar{M R}$ , $\bar{L R}$ , and $\bar{R P}$ are $6$ , $7$ , and $11$ , respectively. What is the length of $\bar{L Q}$ ?

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Explanation

Choice D is correct. The figure shows that angle $M R L$ and angle $P R Q$ are vertical angles. Since vertical angles are congruent, angle $M R L$ and angle $P R Q$ are congruent. It’s given that $\bar{L M}$ is parallel to $\bar{P Q}$ . The figure also shows that $\bar{L Q}$ intersects $\bar{L M}$ and $\bar{P Q}$ . If two parallel segments are intersected by a third segment, alternate interior angles are congruent. Thus, alternate interior angles $M L R$ and $P Q R$ are congruent. Since triangles $L M R$ and $P Q R$ have two pairs of congruent angles, the triangles are similar. Sides $L R$ and $M R$ in triangle $L M R$ correspond to sides $R Q$ and $R P$ , respectively, in triangle $P Q R$ . Since the lengths of corresponding sides in similar triangles are proportional, it follows that $\frac{R Q}{L R} = \frac{R P}{M R}$ . It's given that the lengths of $\bar{M R}$ , $\bar{L R}$ , and $\bar{R P}$ are $6$ , $7$ , and $11$ , respectively. Substituting $6$ for $M R$ , $7$ for $L R$ , and $11$ for $R P$ in the equation $\frac{R Q}{L R} = \frac{R P}{M R}$ yields $\frac{R Q}{7} = \frac{11}{6}$ . Multiplying each side of this equation by $7$ yields $R Q = (\frac{11}{6}) (7)$ , or $R Q = \frac{77}{6}$ . It's given that $\bar{L Q}$ intersects $\bar{M P}$ at point $R$ , so $L Q = L R + R Q$ . Substituting $7$ for $L R$ and $StartFraction 77 Over 6 EndFraction$ for $R Q$ in this equation yields $L Q = 7 + \frac{77}{6}$ , or $L Q = \frac{119}{6}$ . Therefore, the length of $\bar{L Q}$ is $StartFraction 119 Over 6 EndFraction$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the length of $\bar{R Q}$ , not $\bar{L Q}$ .

Choice C is incorrect and may result from conceptual or calculation errors.