Question

The perimeter of an isosceles triangle is $42cm$ and its base is $\frac{3}{2}$ times each of the equal sides. Find the length of each side of the triangle, area of the triangle and height of the triangle.

Answer 1

Let the equal sides of the isosceles triangle each be $x$

Given that length of base $=\frac{3x}{2}$

$\Rightarrow \frac{3x}{2}+x+x=42$cm

$\Rightarrow \frac{3x}{2}+2x=42$

$\Rightarrow 3x+4x=84$

$\therefore x=12$cm

$⟹$ Equal Sides $=12$cm and Base $=18$cm

We know that the height of an isosceles triangle divides it into two right-angled triangles.

$\Rightarrow$ Base${}^2+$ Height${}^2=$ Hypotenuse${}^2$

$\Rightarrow 9^2+h^2={12}^2$

$\Rightarrow h=\sqrt{{12}^2-9^2}$

$h=\sqrt{144-81}$

$h=3\sqrt{7}$cm

Area $=\frac{1}{2}\times$ Base $\times$ Height

$=27\sqrt{7}$ cm${}^2$