Question

Figure shows Δ PQR in which ST || QR and SR and QT intersect each other at M. If `"PT"/"TR" = 5/3` find `("Ar" (triangle "MTS"))/("Ar" (triangle "MQR"))`

Answer 1

Given : `"PT"/"TR" = 5/3`

To find : `("Ar" (triangle "MTS"))/("Ar" (triangle "MQR"))`

Sol : In Δ PST and Δ PRQ

∠ PST = ∠ PQR 

∠ PST = ∠ PQR   ...(Corresponding angles)

∴ Δ PST  ∼  Δ PQR   ....(AA corollary) 

∴ `"PT"/"PR" = "ST"/"QR" = 5/8`   .....(similar sides of similar triangles)

Now, In Δ MTS and  Δ MQR 

∠ MTS = ∠ MQR   ...(Alternate interior angles)

∠ MTS = ∠ MQR

∴ Δ MTS  ∼  Δ MQR   ....(AA corollary)

`therefore ("Ar" (triangle "MTS"))/("Ar" (triangle "MQR")) = "TS"^2/"QR"^2 = (5/8)^2 = 25/64`

i.e. 25 : 64

[The ration of areas of two similar triangle is equal to the ratio of square of their corresponding sides.]