If A is the area of a triangle with sides a,b nd c then find the area of a triangle with sides twice the sides of first triangle

Dear Student,

Please find below the solution to the asked query:

Given : If A is the area of a triangle with sides a , b  and c And sides of another triangle with sides twice the sides of first triangle .

So,

Sides of second triangle  =  2 a , 2 b and 2 c

And

Ratio of their sides :

$\frac{\mathrm{Sides} \mathrm{of} \mathrm{First} \mathrm{triangle}}{\mathrm{Sides} \mathrm{of} \mathrm{Second} \mathrm{triangle}} = \frac{\mathrm{a}}{2 \mathrm{a}} = \frac{\mathrm{b}}{2 \mathrm{b}} = \frac{\mathrm{c}}{2 \mathrm{c}} = \frac{\mathbf{1}}{\mathbf{2}}$ , So

First triangle is similar to second triangle and we know in similar triangles :


$$\begin{gathered} \frac{\mathrm{Area} \mathrm{of} \mathrm{first} \mathrm{triangle} }{\mathrm{Area} \mathrm{of} \mathrm{second} \mathrm{triangle}} = \frac{{\left(\mathrm{Corresponding} \mathrm{side} \right)}^2}{{\left(\mathrm{Corresponding} \mathrm{side} \right)}^2} \\ \Rightarrow \frac{\mathrm{A}}{\mathrm{Area} \mathrm{of} \mathrm{second} \mathrm{triangle}} = \frac{{\mathrm{a}}^2}{{\left( 2\mathrm{a} \right)}^2} \\ \Rightarrow \frac{\mathrm{A}}{\mathrm{Area} \mathrm{of} \mathrm{second} \mathrm{triangle}} = \frac{{\mathrm{a}}^2}{ 4 {\mathrm{a}}^2} \\ \Rightarrow \frac{\mathrm{A}}{\mathrm{Area} \mathrm{of} \mathrm{second} \mathrm{triangle}} = \frac{1}{ 4} \\ \mathbf{\Rightarrow }\mathbf{Area}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{second}\mathbf{ }\mathbf{triangle}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{4}\mathbf{ }\mathbf{A}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$
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