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 adjoining figure, $A B C$ is an isosceles triangle with $A B = A C = 13 c m$ and $B C = 10 c m$ . If $E F G H$ is a square inscribed in $△ A B C$ , find the area of the shaded region.

Accepted Answer

Let, the side of the square be $a$ .

Now, in $△ A B M$

$\Rightarrow A B^{2} = A M^{2} + B M^{2} \Rightarrow {(13)}^{2} = A M^{2} + 5^{2} \Rightarrow 169 - 25 = A M^{2} \Rightarrow \sqrt{144} = A M \Rightarrow 12 = A M$

Thus, $A N = 12 - a$

$△ A B C$ is an isosceles triangle, thus, $A M$ will be perpendicular bisector.

Therefore, $B M = M C = 5 c m, H N = N G = \frac{a}{2}$

Now, $△ A H N$ and $△ A B M$

$\Rightarrow ∠ A H N = ∠ A B M$ (Alternate angles)

$\Rightarrow ∠ A N H = ∠ A M B (90 °)$

$\Rightarrow ∠ B A M = ∠ H A N$ (common angles)

$∴ △ A H N ~ △ A B M$

As we know that if two triangles are similar then the ratio of corresponding sides are equal.

$\Rightarrow \frac{H N}{B M} = \frac{A N}{A M} \Rightarrow \frac{\frac{a}{2}}{5} = \frac{12 - a}{12} \Rightarrow a = \frac{60}{11}$

Area of shaded region

$= a r (△ A B C) - a r (□ E F G H) = \frac{1}{2} \times 12 \times 10 - a^{2} = 60 - {(\frac{60}{11})}^{2} = 30.24 c m^{2}$

Thus, the area of shaded region is $30.24 c m^{2}$

In the adjoining figure, $A B C$ is an isosceles triangle with $A B = A C = 13 c m$ and $B C = 10 c m$ . If $E F G H$ is a square inscribed in $△ A B C$ , find the area of the shaded region.

Important Questions on Triangles

Learn and Prepare for any exam you want !

In the adjoining figure, ABC is an isosceles triangle with AB=AC=13 cm and BC=10cm. If EFGH is a square inscribed in △ABC, find the area of the shaded region.

Important Questions on Triangles

Learn and Prepare for any exam you want !

In the adjoining figure, $A B C$ is an isosceles triangle with $A B = A C = 13 c m$ and $B C = 10 c m$ . If $E F G H$ is a square inscribed in $△ A B C$ , find the area of the shaded region.