Question

Prove that the area of the triangle BCE described on one side BC of a square ABCD as base is one half of the area of similar triangle ACF described on the diagonal AC as base.

Answer 1

ABCD is a square. ∆BCE is described on side BC is similar to ∆ACF described on diagonal AC.

Since ABCD is a square. Therefore,
AB = BC = CD = DA
and AC = `sqrt(2)"BC"` ...[∵ Diagonal = `sqrt(2)` (side)]
Now, ∆BCE ~ ∆ACF
⇒ `"Area (∆BCE)"/"Area (∆ACF)" = "BC"^2/"AC"^2`
⇒ `"Area (∆BCE)"/"Area (∆ACF)" = "BF"^2/(sqrt(2)"BC")^2 = (1)/(2)`
⇒ Area (∆BCE) = `(1)/(2) "area (∆ACF)"`.
Hence proved.