Points \(L, M,\) and, \(N\) have \(xy\)-coordinates \((2,0), (8,12),\) and \((14,0),\) respectively. Points \(P, Q,\) an

[M7MBA]

Points \(L, M,\) and, \(N\) have \(xy\)-coordinates \((2,0), (8,12),\) and \((14,0),\) respectively. Points \(P, Q,\) and \(R\) have \(xy\)-coordinates \((6,0), (8,4),\) and \((10,0),\) respectively. What fraction of the area of the triangle \(LMN\) is the area of the triangle \(PQR?\)

A. \(\dfrac19\)
B. \(\dfrac18\)
C. \(\dfrac16\)
D. \(\dfrac15\)
E. \(\dfrac13\)

[spoiler]OA=A[/spoiler]

Source: GMAT Prep

[Jay@ManhattanReview]

Points \(L, M,\) and, \(N\) have \(xy\)-coordinates \((2,0), (8,12),\) and \((14,0),\) respectively. Points \(P, Q,\) and \(R\) have \(xy\)-coordinates \((6,0), (8,4),\) and \((10,0),\) respectively. What fraction of the area of the triangle \(LMN\) is the area of the triangle \(PQR?\)

A. \(\dfrac19\)
B. \(\dfrac18\)
C. \(\dfrac16\)
D. \(\dfrac15\)
E. \(\dfrac13\)

[spoiler]OA=A[/spoiler]

Source: GMAT Prep

Note that the points L and N lie on X-axis as their y-coordinates are 0. So, the base of the ∆LMN = LM = 14 –2 = 12. We see that the y-coordinate of point M is 12, so, the height of ∆LMN = 12.

Thus, area of ∆LMN = 1/2 of 12*12 = 72

Similarly, note that the points P and R lie on X-axis as their y-coordinates are 0. So, the base of the ∆PQR = PR = 10 – 6 = 4. We see that the y-coordinate of point Q is 4, so, the height of ∆PQR = 4.

Thus, area of ∆PQR = 1/2 of 4*4 = 8

The area of the triangle \(PQR\) 8/72 = 1/9 of the area of the triangle \(LMN\)

Correct answer: A

Hope this helps!

-Jay
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