Name: Areas of Similar Triangles
Uploaded: 2019-07-19
Duration: 6 min 54 s
Description: The corresponding sides of similar triangles are proportional. Are the ratios of areas of similar triangles the same as that of their sides? Watch this video...

Question

In the given figure $M N | | B C$ and $A M : M B = 1 : 2$

find $\frac{area (Δ A M N)}{area (Δ A B C)}$ .

Accepted Answer

We have
$A M : M B = 1 : 2$
$\Rightarrow A M : M B = 1 : 2$

$\Rightarrow \frac{A M}{M B} = \frac{1}{2}$
$\Rightarrow \frac{M B}{A M} = \frac{2}{1}$
Adding $1$ to both sides, we get
$\Rightarrow \frac{M B}{A M} + 1 = \frac{2}{1} + 1$

$\Rightarrow \frac{M B + A M}{A M} = \frac{2 + 1}{1}$

$\Rightarrow \frac{A B}{A M} = \frac{3}{1}$

$\Rightarrow \frac{A M}{A B} = \frac{1}{3}$ .

Now, In $Δ A M N$ and $Δ A B C$
$∠ A M N = ∠ A B C$ (𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 $M N | | B C$ )
$∠ A N M = ∠ A C B$ (𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 $M N | | B C$ )
By $A A$ similarity criterion, $Δ A M N ~ Δ A B C$

If two triangles are similar, then the ratio of their areas is equal to the ratio of the squares oftheir corresponding sides.
$∴ \frac{area (Δ A M N)}{area (Δ A B C)} = {(\frac{A M}{A B})}^{2} = {(\frac{1}{3})}^{2} = \frac{1}{9}$ .

Therefore, the required ratio is $1 : 9$ .

In the given figure $M N | | B C$ and $A M : M B = 1 : 2$

find $\frac{area (Δ A M N)}{area (Δ A B C)}$ .

Important Questions on Triangles

Learn and Prepare for any exam you want !

In the given figure MN||BC and AM:MB=1:2 ​​​​​​​ find area(ΔAMN)area(ΔABC).

Important Questions on Triangles

Learn and Prepare for any exam you want !

In the given figure $M N | | B C$ and $A M : M B = 1 : 2$

find $\frac{area (Δ A M N)}{area (Δ A B C)}$ .