Triangle upper L upper M upper R has a common vertex with triangle upper P upper Q upper R. The

Question

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 $\bar{L Q}$ ?

Accepted Answer

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&rsquo;s given that L M &#175; is parallel to P Q &#175; . The figure also shows that L Q &#175; intersects L M &#175; and P Q &#175; . 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 R Q L R = R P M R . It's given that the lengths of M R &#175; , L R &#175; , and R P &#175; are 6 , 7 , and 11 , respectively. Substituting 6 for M R , 7 for L R , and 11 for R P in the equation R Q L R = R P M R yields R Q 7 = 11 6 . Multiplying each side of this equation by 7 yields R Q = 11 6 7 , or R Q = 77 6 . It's given that L Q &#175; intersects M P &#175; at point R , so L Q = L R + R Q . Substituting 7 for L R and 77 6 for R Q in this equation yields L Q = 7 + 77 6 , or L Q = 119 6 . Therefore, the length of L Q &#175; is 119 6 . Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the length of R Q &#175; , not L Q &#175; . Choice C is incorrect and may result from conceptual or calculation errors.

Question 9f3f17