Question

The sides of a triangle are $120m,170m,250m.$ Find its area and height of the triangle if the base is $250m.$

Answer 1

We have, Sides of a triangle are $120m,170m,250m$

Base of the given triangle $=250m$

Let us assume the sides as

$a=120m,b=170m,c=250m$

s $=$ semi-perimeter $=\frac{a+b+c}{2}$

$=\frac{(120+170+250)}{2}$
$=\frac{540}{2}$
$=270$

Applying Heron's Formula to find the area of the given triangle

Area of the given triangle $=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{270(270-120)(270-170)(270-250)}$
$=\sqrt{270\times 150\times 100\times 20}$
$=\sqrt{(3\times 3\times 3\times 10)\times (3\times 5\times 10)\times (10\times 10)\times (2\times 2\times 5)}$

$=\sqrt{\underset{–}{3\times 3}\times \underset{–}{3\times 3}\times \underset{–}{10\times 10}\times \underset{–}{10\times 10}\times \underset{–}{2\times 2}\times \underset{–}{5\times 5}}$

$=3\times 3\times 10\times 10\times 2\times 5$

$=9\times 10\times 10\times 10$

$=9000$
Therefore area of the triangle is $9000m^2$

We know that area of a triangle can be given by the formula
Area of triangle $=\frac{1}{2}\times base\times height$
$\therefore 9000=\frac{1}{2}\times 250\times$ height [$\because$ area of triangle $=9000m^2$]
$\Rightarrow 9000=125\times height$
$\Rightarrow Height=\frac{9000}{125}$
$\Rightarrow Height=72m$