Question

In the given figure, DE || BC
(i) If DE = 4 cm, BC = 6 cm and Area (∆ADE) = 16 cm², find the area of ∆ABC.
(ii) If DE = 4 cm, BC = 8 cm and Area (∆ADE) = 25 cm², find the area of ∆ABC.
(iii) If DE : BC = 3 : 5. Calculate the ratio of the areas of ∆ADE and the trapezium BCED.

Answer 1

SOLUTION :  

GIVEN : DE || BC.

In ΔADE and ΔABC

∠ADE =∠B     (Corresponding angles)

∠DAE =∠BAC   (Common)

ΔADE ∼ ΔABC   (By AA Similarity)

(i) Given : DE = 4m, BC = 6 cm and Area (ΔADE) =16cm²

We know that the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

Hence,

ar(ΔADE)/ar(ΔABC) = DE²/BC²

16/ar(ΔABC) = 4²/6²

16/ar(ΔABC) = 16/36

ar(ΔABC) =( 36 × 16)/16

ar(ΔABC) = 36 cm²

Hence, ar(ΔABC) = 36 cm²

 

(ii) GIVEN : DE || BC , DE = 4 cm, BC = 8 cm and Area (ΔADE) = 25cm²

In ΔADE and ΔABC

∠ADE =∠B     (Corresponding angles)

∠DAE =∠BAC   (Common)

ΔADE ∼ ΔABC   (By AA Similarity)

We know that the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

Hence,

ar(ΔADE)/ar(ΔABC) = DE²/BC²

25/ar(ΔABC) = 4² / 8²

25/ar(ΔABC) = 16 / 64

ar(ΔABC) = (64 × 25)/16

ar(ΔABC) = 4 × 25

ar(ΔABC) = 100 cm²

Hence, ar(ΔABC) = 100 cm²

 

(iii) Given :  DE : BC = 3 : 5.

In ΔADE and ΔABC

∠ADE =∠B     (Corresponding angles)

∠DAE =∠BAC  (Common)

ΔADE ∼ ΔABC   (By AA Similarity)

We know that the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

Hence,

ar(ΔADE)/ar(ΔABC) = DE²/BC²

ar(ΔADE)/ar(ΔABC) = 3² / 5²

ar(ΔADE)/ar(ΔABC) = 9/ 25

Let area of ΔADE = 9x sq units and area of ΔABC = 25x sq units.

ar (trap.BCED) = ar(∆ABC) - ar(∆ADE)

ar (trap.BCED) = 25x - 9x = 16x sq units.

ar (trap.BCED) = 16x sq units.

Now, Ar(ΔADE) / ar(trapBCED) = 9x/16x

ar(ΔADE)/ Ar(trapBCED) = 9/16

Hence, the ratio of the areas of ΔADE and the trapezium BCED is ar(ΔADE) : Ar(trapBCED) = 9 :16.

HOPE THIS ANSWER WILL HEYOU....

Answer 2

(i) Triangle ABC and triangle ADE are similar triangles. Hence, their ratios will be same.

Ratio of DE to BC is 4:6 => 2:3

To find area, we need to raise the ratio of length to power 2

(2/3)² = 16/x

4/9 = 16/x

x = (16*9)/4

x = 36cm²

Therefore, area of triangle ABC is 36cm²

(ii) Ratio of DE to BC is 4:8 => 1:2

(1/2)² = 25/x

1/4 = 25/x

x = 25*4

x = 100cm²

Therefore, area of triangle ABC is 100cm²

I'm not sure about the third question, sorry