Question

In the following figure, point D divides AB in the ratio 3 : 5. Find :

1. `(AE)/(EC)`
2. `(AD)/(AB)`
3. `(AE)/(AC)`
   Also, if:
4. DE = 2.4 cm, find the length of BC.
5. BC = 4.8 cm, find the length of DE.

Answer 1

i. Given that `(AD)/(DB) = 3/5`

Now, DE is parallel to BC.

Then, by basic proportionality theorem, we have

`(AD)/(DB) = (AE)/(EC)`

`=> (AE)/(EC) = 3/5`

ii. Given that `(AD)/(DB) = 3/5`

So, `(AD)/(AB) = 3/8`

iii. Given that `(AD)/(DB) = 3/5`

So, `(AD)/(AB) = 3/8`

In ΔADE and ΔABC,

∠ADE = ∠ABC  ...(Since DE || BC, so the angles are corresponding angles)

∠A = ∠A   ...(Common angle)

∴ ΔADE ∼ ΔABC  ...(AA criterion for similarity)

`=> (AD)/(AB) = (AE)/(AC)`

`=> (AE)/(AC) = 3/8`

iv. Given that `(AD)/(DB) = 3/5`

So, `(AD)/(AB) = 3/8`

In ΔADE and ΔABC,

∠ADE = ∠ABC  ...(Since DE || BC, so the angles are corresponding angles)

∠A = ∠A    ...(Common angle)

∴ ΔADE ∼ ΔABC  ...(AA criterion for similarity)

`=> (AD)/(AB) = (DE)/(BC)`

`=> 3/8 = (2.4)/(BC)`

`=>` BC = 6.4 cm

v. Given that `(AD)/(DB) = 3/5`

So, `(AD)/(AB) = 3/8`

In ΔADE and ΔABC,

∠ADE = ∠ABC  ...(Since DE || BC, so the angles are corresponding angles)

∠A = ∠A  ...(Common angle)

∴ ΔADE ∼ ΔABC  ...(AA criterion for similarity)

`=> (AD)/(AB) = (DE)/(BC)`

`=> 3/8 = (DE)/(4.8)`

`=>` DE = 1.8 cm