Question

If two triangles ABC and DEF are equiangular, AL and DM are perpendicular from A and D on opposite sides BC and EF prove that BC/EF=AL/DM

Answer 1

Solution:

As, given Δ ABC and ΔDEF are equiangular.

The meaning of Equiangular is that corresponding angles of two triangles are same.

Means, ∠A=∠D,

∠B=∠E

∠C=∠F

So, Δ ABC ~ ΔDEF→→[AA similarity criterion]

Now, AL and DM are perpendicular from A and D on opposite sides BC and E F.

So, when triangles are similar, their sides are proportional.

$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$-----(1)

Now, In ΔALB and ΔDME

∠ALB=∠DME=90°

∠B=∠E→→Given

ΔALB ~ ΔD ME→→→→→[AA]

→when triangles are similar, their sides are proportional.

$\frac{AB}{DE}=\frac{AL}{DM}$-------(2)

From (1) and (2)

$\frac{CB}{FE}=\frac{AL}{DM}$