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Question

Solve the following questions:
In $$\triangle ABC$$ and $$DE\parallel BC$$ and $$AD:DB=2:3$$ then find the ratio of areas of $$\triangle ADE$$ and $$\triangle ABC$$

Solution
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In $$\triangle ABC$$ $$BC\parallel DE$$ and $$\cfrac{AD}{DB}=\cfrac{2}{3}$$ (given)
In $$\triangle ABC$$ and $$\triangle DEA$$
$$\angle B=\angle D$$
$$\angle C=\angle E$$
AA similarity cirterion
$$\cfrac { ar.\left( \triangle ADE \right) }{ ar.\left( \triangle ABC \right) } =\cfrac { { \left( AD \right) }^{ 2 } }{ { \left( AB \right) }^{ 2 } } =\cfrac { { \left( AD \right) }^{ 2 } }{ { \left( AD+DB \right) }^{ 2 } } $$
$$=\cfrac { { \left( \cfrac { AD }{ BD } \right) }^{ 2 } }{ { \left( \cfrac { AD }{ BD } +1 \right) }^{ 2 } } ={ \left( \cfrac { AD }{ BD } \right) }^{ 2 }\times \cfrac { 1 }{ { \left( \cfrac { AD }{ BD } +1 \right) }^{ 2 } } ={ \left( \cfrac { 2 }{ 3 } \right) }^{ 2 }\times \cfrac { 1 }{ { \left( \cfrac { 2 }{ 3 } +1 \right) }^{ 2 } } =\cfrac { 4 }{ 9 } \times \cfrac { 9 }{ 25 } =\cfrac { 4 }{ 25 } $$

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Similar Questions
Q1
Solve the following questions:
In $$\triangle ABC$$ and $$DE\parallel BC$$ and $$AD:DB=2:3$$ then find the ratio of areas of $$\triangle ADE$$ and $$\triangle ABC$$
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Q2
In the figure, BCDE and ABAD=34
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534279_19db78501c2846849c7c7129ad9c2611.png
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Q3
In the figure given below, DEBC and AD:DB=1:2, find the ratio of the areas of ADE and trapezium DBCE.
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Q4
In the given figure, DE || BC
(i) If DE = 4 cm, BC = 6 cm and Area (āˆ†ADE) = 16 cm2, find the area of āˆ†ABC.
(ii) If DE = 4 cm, BC = 8 cm and Area (āˆ†ADE) = 25 cm2, find the area of āˆ†ABC.
(iii) If DE : BC = 3 : 5. Calculate the ratio of the areas of āˆ†ADE and the trapezium BCED.

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Q5

In Fig. 7.179, DE || BC
(i)If DE= 4 cm, BC = 6cm and Area (Ī”ADE)=16cm2, find the area of Ī”ABC.
(ii) If DE =4 cm, BC = 8 cm and Area (Ī”ADE)=25cm2, find the area of Ī”ABC.
(iii) If DE : BC = 3 : 5. Calculate the ratio of the areas of Ī”ADE and the trapezium BCED.

Fig. \( 7.179 \)


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