Question

Find the area of an isosceles triangle whose one side is 10 cm greater than each of its equal sides and perimeter is 100 cm.

Answer 1

$\underline{\underline{\bold{Answer}}}$


446 ${cm}^{2}$


$\huge\bold{Explanation}$


$\underline{\underline{Given}}$


perimeter = 100cm


Third side = greater than it's equal side by 10cm. = x + 10


$\underline{\underline{To\:find}}$


Area of the triangle


$\underline{\bold{According\:to\:the\:question}}$


Let the two equal side of the isosceles triangle be x and x which is 2x.


$\underline{\bold{We\:know\:that}}$


$\boxed{\bold{Perimeter =sum\:of\:all\:sides}}$


$\implies{2x + x + 10 = 100}$


Add 2x and x...


$\implies{3x + 10 = 100}$


Let subtract the 10 from 100...


$\implies{3x = 90}$


Let shift 3 to Right hand side...


$\implies{x = \dfrac{90}{3}}$


after cancelling...


$\implies{x = 30cm}$


Let find the measure of all sides...


First side = x = 30cm


Second side = x = 30cm


Third side = x + 10 = 30 + 10cm = 40cm


Now , we have to find the area of the isosceles triangle.....


Let use the heron's formula to find the area.


$\bold{Heron's\:Formula}$



$\boxed{\boxed{\bold{Area=\sqrt{P(P - a)(P - b)(P-c)}}}}$


$\implies{\sqrt{50\times20\times 20\times 10}}$


$\implies{446{cm}^{2}}$

Answer 2

$\large\bold{200 \sqrt{5}}$

$\large\bold\red{Explanation}$

let the one of the equal side be x cm.
➡ measure of greater side: (x+10) cm.

Hence,
perimeter of the triangle = (x + x + x+10) = (3x +10) cm. ----(1)

Given, perimeter = 100 cm. ----(2)

from (1) and (2), we get:

➡ 3x + 10 = 100

➡ 3x = 100 - 10

➡ 3x = 90

➡ x = 90/3

➡ x = 30

Hence,

equal sides of triangle: 30 cm.
unequal side: 40 cm.

it's height: $\sqrt{ {(30) }^{2} - {(20)}^{2} } = \sqrt{900 - 400} = \sqrt{500}$

➡ $10\sqrt{5}$ = Height of the triangle.

➡ Base of the triangle = 40 cm. [given]

we know that :

area of a triangle = $\large{\frac{1}{2} \times base\times height }$

➡ it's area = $\large{\frac{1}{2} \times 40 \times 10 \sqrt{5}}$

➡ $\large\bold{200 \sqrt{5} }$

Hence, area of the triangle = $\large\bold\red{200 \sqrt{5} }$