Question

The ratio of the measure of the three sides of a traingle is 4:2:3.and it's perimeter is 36 cm.Find the area of this traingle​

Answer 1

Given: Ratio of sides of a triangle is 4:2:3 & Perimeter of park is 36 cm.

To find: Area of triangle?

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☯ Let the sides of triangle a, b and c be 4x, 2x and 3x respectively.

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$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

$\star\;{\boxed{\sf{\pink{Perimeter_{\;(triangle)} = Sum\:of\:\:it's\:all\:sides}}}}$

$:\implies\sf a + b + c = 36\\ \\ \\ :\implies\sf 4x + 2x + 3x = 36\\ \\ \\ :\implies\sf 9x = 36\\ \\ \\ :\implies\sf x = \cancel{ \dfrac{36}{9}}\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{x = 4}}}}}\;\bigstar$

Therefore, Sides of triangle are,

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• a = 4 × 4 = 16 cm
• b = 2 × 4 = 8 cm
• c = 3 × 4 = 12 cm

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$\underline{\bigstar\:\boldsymbol{Using\:Herons\:Formula\::}}$

$\star\;{\boxed{\sf{\pink{Area_{\;(triangle)} = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

Where,

$:\implies\sf s = semi - perimeter$

$:\implies\sf s = \dfrac{a + b + c}{2}\\ \\ \\ :\implies\sf s = \dfrac{16 + 8 + 12}{2}\\ \\ \\ :\implies\sf s = \cancel{\dfrac{36}{2}}\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{s = 18\:cm}}}}}\;\bigstar$

Now,

$\sf We\:have \begin{cases} & \sf{a = \bf{16\:cm}} \\ & \sf{b = \bf{8\:cm}} \\ & \sf{c = \bf{12\:cm}} \\ & \sf{s = \bf{18\:cm}} \end{cases}$

$\dag\;{\underline{\frak{Putting\:values\:in\;formula,}}}$

$:\implies\sf Area_{\;(triangle)} = \sqrt{18(18 - 16)(18 - 8)(18 - 12)}\\ \\ \\ :\implies\sf Area_{\;(triangle)} = \sqrt{18 \times 2\times 10 \times 6}\\ \\ \\ :\implies\sf Area_{\;(triangle)} = \sqrt{(3)^2 \times (2)^2 \times 10 \times 6}\\ \\ \\ :\implies\sf Area_{\;(triangle)} = 3 \times 2 \sqrt{60}\\ \\ \\ :\implies\sf Area_{\;(triangle)} = 6 \sqrt{60}\\ \\ \\:\implies{\underline{\boxed{\frak{\purple{Area_{\;(triangle)} = 46.48\:cm^2}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Area\:of\:triangle\:is\: \bf{46.48\:cm^2}.}}}$

Answer 2

Answer:

• Area of triangle is 46.44 cm².

Step-by-step explanation:

Given :-

• Ratio of measure of three sides of triangle is 4:2:3.
• Perimeter of triangle is 36 cm.

To find :-

• Area of triangle.

Solution :-

Let, Sides of triangle be 4x, 2x and 3x.

Perimeter of triangle = Sum of all sides

$\longrightarrow$ 4x + 2x + 3x = 36

$\longrightarrow$ 9x = 36

$\longrightarrow$ x = 36/9

$\longrightarrow$ x = 4

Sides :-

• 4x = 4×4 = 16

• 2x = 2×4 = 8

• 3x = 3×4 = 12

Sides of triangle are 16 cm, 8 cm and 12 cm.

Now,

Here, We do not have height of triangle.

We will use Heron's formula that is :

Area of triangle = √s(s - a)(s - b)(s - c)

Where,

• a, b and c are sides of triangle.
• s is semi-perimeter of triangle.

So,

Semi-perimeter = Perimeter/2

$\longrightarrow$ Semi-perimeter = 36/2

$\longrightarrow$ Semi-perimeter = 18

Semi-perimeter of triangle is 18 cm.

Put all values in area formula :

$\longrightarrow$ Area = √18 × (18 - 16)(18 - 8)(18 - 12)

$\longrightarrow$ Area = √18 × 2 × 10 × 6

$\longrightarrow$ Area = √2 × 3 × 3 × 2 × 2 × 5 × 2 × 3

$\longrightarrow$ Area = 2 × 2 × 3 √5 × 3

$\longrightarrow$ Area = 12 × √15

$\longrightarrow$ Area = 12 × 3.87

$\longrightarrow$ Area = 46.44

Therefore,

Area of triangle is 46.44 cm².