Question

In Δ ABC , MN || BC .

If `"AB"/"AM" = 9/4` , find `("Ar" ("trapezium MBCN"))/("Ar" . (triangle "ABC"))`

Answer 1

Δ AMN  ∼  ΔABC  {proved above}

`("Ar" triangle  "AMN")/("Ar"  triangle "ABC") = "AM"^2/"AB"^2 = (4/9)^2 = 16/81` 

`("Ar"  triangle "ABC" - "Ar" ("trapezium MBCN"))/("Ar" . triangle "ABC") = 16/81`

81 ×  Ar. Δ ABC - 81 × Ar. (trapezium MBCN) = 16 × Ar. Δ ABC

64 Ar. Δ ABC = 81 Ar .(trapezium MBCN)

`("Ar" ("trapezium MBCN"))/("Ar".(triangle "ABC")) = 65/81`

i.e. 65 : 81