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03 May 2020, 10:57
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In the figure above, M and N are two points on the side PQ and PR of a triangle PQR, respectively such that MNQR is a trapezium and MN:QR = 2:5. What is the ratio of the area of triangle PMN to the area of trapezium MNQR?
A. 4:25
B. 4:21
C. 4:10
D. 4:9
E. 1:2
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Attachment:
Untitled.png [ 5.7 KiB | Viewed 19112 times ]
Bunuel
Given: In the figure above, M and N are two points on the side PQ and PR of a triangle PQR, respectively such that MNQR is a trapezium and MN:QR = 2:5.
Asked: What is the ratio of the area of triangle PMN to the area of trapezium MNQR?
Attachment:
Screenshot 2020-05-03 at 11.42.06 PM.png [ 21.27 KiB | Viewed 15700 times ]
Let PLO be perpendicular drawn from P to line QR meeting line QR at O and line MN at L.
If MNQR is a trapezium -> MN & QR are parallel
Since triangles PMN & PQR are similar triangles
MN:QR = PL:PO = 2:5
Ratio of area of triangle PMN to the area of triangle PQR = MN * PL : QR : PO = 2^2 : 5^2 = 4:25
Since area of trapezium MNQR = Area of triangle PQR - Area of triangle PMN
Ratio of the area of triangle PMN to the area of trapezium MNQR = 4:(25-4) = 4: 21
IMO B
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03 May 2020, 11:12
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If MNQR is a trapezium, then MN||QR which means PMN and PQR are similar triangles
By property, if the corresponding sides of two similar triangles are in the ratio \(a:b\), then their areas are in the ratio \(a^2:b^2\)
So the areas of PMN and PQR are in the ratio 4:25 and therefore the area of MNQR and PQR must be in the ratio 21:25
Therefore PMN and MNQR have their areas in the ratio 4:21
Answer is (B)
Posted from my mobile device
By property, if the corresponding sides of two similar triangles are in the ratio \(a:b\), then their areas are in the ratio \(a^2:b^2\)
So the areas of PMN and PQR are in the ratio 4:25 and therefore the area of MNQR and PQR must be in the ratio 21:25
Therefore PMN and MNQR have their areas in the ratio 4:21
Answer is (B)
Posted from my mobile device
