Question

6.
The corresponding sides of two similar triangles are in the ratio 1:4, then
the ratios of their areas is
(A) 1:2
(B) 1:16
(C)
1:4
(D) 16:1.
​

Answer 1

Working Out:

The above question is asking about the ratio of areas of triangles whose corresponding sides' ratio is provided. For this, we should know about the relation between Area and corresponding sides of similar triangles.

• Given, Ratio of corresponding sides = 1:4

The theoram says that, area of two similar triangles are proportional to the squares of their corresponding sides. So, let's represent this in a proportional way.

➝ $\Large{\rm{ \frac{Area \: of \: \triangle \: A}{Area \: of \: \triangle \: B} = \frac{ (Corresponding \: side \: of \: A) {}^{2} }{(Corresponding \: side \: of \: B) {}^{2} }} }$

• Let the sides of the triangle be x and 4x

Now plug the ratio of corresponding sides in the above relation based on the theoram:

➝ $\Large{\rm{ \frac{Area \: of \: 1st \: \triangle}{Area \: of \: 2nd \: \triangle} = \frac{x {}^{2} }{ {(4x)}^{2} }} }$

➝ $\Large{\rm{ \frac{Area \: of \: 1st \: \triangle}{ Area \: of \: 2nd \triangle} = \frac{ {x}^{2} }{16 {x}^{2} } }}$

Cancel x² from NUM. and DEN. (LHS)

➝ $\Large{ \rm{ \frac{Area \: of \: 1st \: \triangle}{Area \: of \: 2nd \: \triangle} = \frac{1}{16} }}$

So, the required ratio: 1 : 16 (Option B)

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Answer 2

Step-by-step explanation:

CASE:

THE CORRESPONDING SIDES OF TWO SIMILIAR TRIANGLES ARE IN THE RATIO 1:4 .

TO FIND:

RATIO OF THEIR AREAS.

THEOREM USED:

AREA OF TWO SIMILIAR TRIANGLES IS PROPORTIONAL TO SQUARES OF THEIR CORRESPONDING SIDES.

SOLUTION:

LET THE CORRESPONDING SIDES BE ,

X AND 4X.

$\frac{area \: of \: triangle \: a}{area \: of \: triangle \: b} \\ \\ = > \frac{ {x}^{2} }{ ({4x})^{2} } = \frac{ {x}^{2} }{ {16x}^{2} } = \frac{1}{16}$

HENCE THE OF THE AREAS IS 1:16.