Question

@12. An isosceles right-angled triangle has an area 8 cm2. The value of perimeter of triangle is ..... cm.​

Answer 1

Answer :-

13.656 cm

Solution :-

Let the equal sides of the isosceles right triangle be 'a'cm

Area of isosceles right triangle = 8 cm²

==> 1/2 * a² = 8

==> a² = 8 * 2

==> a² = 16

==> a = √16

==> a = 4

Therefore the equal sides are of length 4 cm.

Let the hypotenuse of the isosceles right triangle be h

By pythagoars theorem

==> h² = a² + a²

==> h² = 2a²

==> h² = 2( 4 )²

==> h = 4√2

==> h = 4( 1.414 )

==> h = 5.656

Perimeter of the triangle = Sum of 3 sides

= a + a + h

= 2a + h

= 2( 4 ) + 5.656

= 8 + 5.656

= 13.656 cm

Hence the perimeter of the triangle is 13.656.

Answer 2

Solution :

Let ABC be an isosceles right-angled triangle in which AB = BC

Given Area of Triangle = 8 cm^2

$\boxed{ \purple{ \sf \: Area \: of \: Triangle = \frac{1}{2} \times base \times height}}$

$\longrightarrow \sf \dfrac{1}{2} \times AB \times BC = 8 {cm}^{2}$

Since It's an Isosceles Triangle Two Sides are Equal.

$\longrightarrow \sf \dfrac{1}{2} \times {AB}^{2} = 8 {cm}^{2}$

$\longrightarrow \sf {AB}^{2} = 8 \times 2 {cm}^{2}$

$\longrightarrow \sf {AB}^{2} =16 {cm}^{2}$

$\longrightarrow \sf {AB} = \sqrt{16}$

$\longrightarrow \sf {AB} = 4 \: cm$

We've got measurement of Base and Perpendicular,In order to find the Perimeter We've to Find the Hypotenuse First.

$\bf \underline{ \sf \: By \: Pythagoras \: Theorem : }$

$\boxed{ \purple{ \sf \: ({Hypotenuse})^{2} = ({Perpendicular})^{2} + ( {Base })^{2} }}$

$\longrightarrow \sf \: {AC}^{2} = {AB}^{2} + {BC}^{2}$

$\longrightarrow \sf \: {AC}^{2} = {4}^{2} + {4}^{2}$

$\longrightarrow \sf \: {AC}^{2} = 16 + 16$

$\longrightarrow \sf \: {AC}^{2} = 32$

$\longrightarrow \sf \: {AC} = \sqrt{32}$

$\longrightarrow \sf \: {AC} = 5.65 \: cm$

Now,

$\boxed{ \purple{ \sf \: Perimeter \: of \: Triangle \: = Sum \: of \: all \: Sides }}$

$\longrightarrow \sf \: AB + BC + AC$

$\longrightarrow \sf \:4 \: cm + 4 \: cm + 5.65 \: cm$

$\longrightarrow \sf \:13.65 \: cm$

Hence,Value of Perimeter of the Triangle is 13.65 cm. (Approx)