Question

The sides of triangle in ratio is 2:3:4 and the perimeter is 90 find it's area?

Answer 1

                              $\huge{\boxed{Solution}}$.

$\bold{Given:}$

• Sides of triangle in ratio is 2:3:4
• The perimeter is 90 cm

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$\bold{To\:Find:}$

• Area of triangle

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$\bold{Let's\:solve,}$

Let the sides of triangle is

$2x,\:3x\:and\:4x$

We know that perimeter

is equal to Sum of all sides

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Perimeter = Sum of all sides

⇒ 90 = $2x+3x+4x$

⇒ 90 = 9$x$

$\therefore x=\dfrac{90}{9}=10$

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So,

Side $a$ = 2$x$ = 2 × 10 = 20 $cm$

Side $b$ = 3$x$ = 3 × 10 = 30 $cm$

Side $c$ = 4$x$ = 4 × 10 = 40 $cm$

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Here we use Heron's formula

to find area of Δ

So, firstly we have to find

semi perimeter

$\implies S=\dfrac{20+30+40}{2}\\\\\implies S=\dfrac{90}{2}\\\\\huge\boxed{{\therefore S=42\:cm}}$

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$\huge{\boxed{{Area=\sqrt{S(S-a)(S-b)(S-c)}}}}$

$\bold{Area=\sqrt{42(42-20)(42-30)(42-40)}}$

$\bold{Area=\sqrt{42\times(22)\times(12)\times(2)}}$

$\bold{Area=\sqrt{2\times3\times7\times2\times11\times2\times2\times3\times2}}$

$\bold{Area=2\times2\times3\sqrt{2\times7\times11}}$

$\bold{Area=12\sqrt{154}}$