Question

triangle XYZ and triangle PQR are two equilateral triangles such that Q is the mid point of XY .Prove that the ratio of area of triangle XYZ to the area of triangle PQR is 4:1.with rough diagram

Answer 1

ar of XYZ/ ar of PQR = √3/4 XY^2/ √3/4 PQ^2
= XY^2 / (1/2 XY)^2. As PQ = 1/2 XY
= 4/1....= 4:1

Answer 2

∆PQR and ∆XYZ both are equilateral triangle . in which Q is the midpoint of XY .
in the same way , R ia the midpoint of YZ and
P is the midpoint of ZX.
in this way,
you see P, and R is the midpoint of side ZX and and ZY ,
according to reverse of Thales theorem ,
PR || XY ,

so, PR/XY = ZP/ZX =ZR/ZY =1/2
so, PQ = PR = RQ =XY/2
now,

∆XYZ and ∆PQR
we know both are equilateral triangle,
so, this is always similar

∆XYZ ~∆PQR

we also know,
ratio of area of similar triangle are square of ratio of corresponding side of traingle .
if two triangle ∆XYZ and ∆PQR are similar , then
ar∆XYZ/ar∆PQR =(XY/PQ)^2 =(YZ/QR)^2 =(ZX/RP)^2

we know,
XY/2 = PQ

use this , in above theorem ,

ar∆XYZ/ar∆PQR = (XY/PQ)^2

=(XY/XY/2)^2

=(2/1)^2
=4/1

hence,

ar∆XYX : ar∆PQR = 4 : 1