Question

In the given figure DE || BC .prove that ∆ADE, and ∆ABC are similar.given that AD =1/2 BD, calculate DE, if BE = 4.5cm.
Also find Area ∆ADE/Area ∆ABC
and
Area ∆ADE/Area Trapezium BCED​

Answer 1

Given :  DE || BC

AD =1/2 BD,

BC = 4.5cm

To Find :   prove that ∆ADE, and ∆ABC are similar.

calculate DE

Area ∆ADE/Area ∆ABC

 Area ∆ADE/Area Trapezium BCED​

Solution:

in ΔADE & ΔABC

∠A = ∠A   common

∠D = ∠B   ( corresponding angles)

∠E = ∠C   ( corresponding angles)

=> ΔADE ≈ ΔABC    (AAA)

QED

Hence proved

AD/AB  =  DE/BC

=> AD/(AD + DB)    = DE/4.5

AD =1/2 BD, => BD = 2AD

=> AD/(AD + 2AD)  = DE/4.5

=> AD/3AD = DE/4.5

=> DE = 1.5   cm

AD/AB  =  DE/BC = 1/3

and AB/AD = BC/DE = 3

ratio of area of similar triangle = ( Ratio of corresponding sides)²

Area of  ΔABC = (3)² area of Δ ADE

=> Area of  ΔABC = 9 area of Δ ADE

area of Δ ADE /Area of  ΔABC =  1/9

Area Trapezium BCED​  = Area of  ΔABC -  area of Δ ADE

=> Area Trapezium BCED​  = 9 area of Δ ADE -  area of Δ ADE

=> Area Trapezium BCED​  = 8 area of Δ ADE

=> area of Δ ADE / Area Trapezium BCED​  = 1/8

Learn more:

In triangle ABC, seg DE || SEG BC. if twice area of triangle ADE ...

brainly.in/question/4599623

In the given figure, If DE || BC and AD : DB = 1:2 then Area of ∆ADE ...

brainly.in/question/17155687