O is a point inside the triangle MNP.

UV is a line drawn through O parallel to MP which intersects MN at U and NP at V.

RQ is a line drawn through O parallel to MN which intersects MP at R and NP at Q.

ST is a line drawn through O parallel to NP which intersects MN at S and MP at T.

The respective areas of triangles ROT, UOS and QOV are 4, 9 and 49

Find the area of triangle MNP

MNP, ROT, SOU, and QOV are similar triangles. MUOR, NQOS, and PVOT are parallelograms.

Corresponding sides of similar triangles are proportionate to the square root of the areas of the triangles. In particular RO:OQ = 2/7 and RO:US = 2:3.

Opposite sides of a parallelogram are congruent. In particular RO=MU and OQ=SN.

Combining these relations yields MU:US = 2:3 and MU:SN = 2:7. From this MU:MN = RO:(MU+US+SN) = 2:(2+3+7) = 2:12. Then the ratio of areas of ROT:MNP is 4:144.

The area of MNP is 144.

In general: sqrt[area(MNP)] = sqrt[area(ROT)] + sqrt[area(SOU)] + sqrt[area(QOV)]