PQR is a triangle. MN is a line segment intersecting PQ in M and PR in N such that MN ∥ QR and divides ΔPQR into two parts, which are equal in area. What is the ratio of MQ to PQ?
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e. .
Area of ΔPMN = Area of quadrilateral MNQR
Since MN divides ΔPQR into two equal areas, we know that the ratio of their areas is 1:1.
Area of ΔPMN/Area of ΔPQR = 1/2
The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides.
(Area of ΔPMN/Area of ΔPQR) = (PM/PQ)2 = 1/2
⇒ PM/PQ = 1/√2
⇒ MQ/PQ = 1 - (1/√2) = (√2 - 1)/√2
Therefore, the ratio of MQ to PQ is (√2 - 1) : √2.