Question

In ΔXYZ ∽ ΔDEF consider the correspondence XYZ ⇔ EDF. If Perimeter of ΔXYZ/Perimeter of ΔDEF =3/4, Find XY/ED and XZ+YZ/EF+DF

Answer 1

Given, ΔXYZ ∽ ΔDEF consider the correspondence XYZ ⇔ EDF.

Hence, $\frac{XY}{ED}=\frac{YZ}{DF}=\frac{XZ}{EF}$

so, $\frac{XY}{ED}=\frac{YZ}{DF}=\frac{XZ}{EF}=\frac{XY+YZ+XZ}{ED+DF+EF}$-----(1)

Given, $\frac{\text{perimeter of XYZ}}{\text{perimeter of ABC}}=\frac{3}{4}$

e.g., $\frac{XY+YZ+XZ}{ED+DF+EF}=\frac{3}{4}$-----(2)

put equation (1) in equation (2),
then, $\frac{XY}{ED}=\frac{YZ}{DF}=\frac{XY+YZ+XZ}{ED+DF+EF}=\frac{3}{4}$ ----(3)

again, $\frac{XY}{ED}=\frac{YZ}{DF}=\frac{XZ}{EF}$

$\frac{XZ}{EF}=\frac{YZ}{DF}=\frac{XZ+YZ}{EF+DF}\\\\\implies\frac{XZ+YZ}{EF+DF}=\frac{3}{4}$

[from equation (3), .. ]

hence, XY/ED = 3/4
and (XZ + YZ)/(EF + DF) = 3/4

Answer 2

Dear student,

Solution:In ΔXYZ ∽ ΔDEF consider the correspondence XYZ ⇔ EDF

according to the correspondence

XY/ED = YZ/DF = ZX/FE

If Perimeter of ΔXYZ/Perimeter of ΔDEF =3/4

Perimeter of ΔXYZ = XY+YZ+ZX

Perimeter of ΔDEF = ED+DF+FE

Ratio of both perimeter is given as 3/4

$\frac{XY+YZ+ZX}{ED+DF+FE} = \frac{3}{4}$

If ratio of these sides are equal then we can write

XY/ED = YZ/DF = ZX/FE =(XY+YZ+ZX)/(ED+DF+FE)

From this expression we can write that all thr ratios of length of sides are equal to 3/4

so, XY/ED = 3/4

YZ/DF = 3/4

ZX/FE = 3/4

by this way we can write that addition of any two sides in numerator of triangle XYZ by addition of any two sides of triangle DEF in denominator gives 3/4

so,

XZ+YZ/EF+DF = 3/4

Hope it helps you.