Question

The Perimeter of an Isosceles triangle is 42 cm. The ratio of the equal side to its base is 3: 4. Find the area of the triangle.

Answer 1

Answer:

Area of triangle is 78.893 cm².

Step-by-step explanation:

Given that,

Perimeter of an isosceles triangle is 42 cm.

Ratio of the equal side to its base is 3:4.

Let,

Two sides of triangle be 3x and 4x

And, Third side of triangle be 3x [Given triangle is isosceles triangle and we have ratio of one equal side and base of equal side. So, the two equal sides will be 3x and 3x].

Then,

Perimeter of triangle = Sum of all sides

→ 3x + 3x + 4x = 42

→ 10x = 42

→ x = 42/10

→ x = 4.2

Sides :

• 3x = 3 × 4.2 = 12.6 cm.
• 3x = 3 × 4.2 = 12.6 cm.
• 4x = 4 × 4.2 = 16.8 cm.

We know,

Heron's formula :

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

[Where s is semi-perimeter, a, b and c is sides of triangle]

So,

• Semi-perimeter = Perimeter/2

→ Semi-perimeter = 42/2

→ Semi-perimeter = 21

Semi-perimeter is 21 cm.

Now,

→ Area = √21(21 - 12.6)(21 - 12.6)(21 - 16.8)

→ Area = √21 × 8.4 × 8.4 × 4.2

→ Area = 8.4 × √88.2

→ Area = 8.4 × 9.392

→ Area ≈ 78.893

∴ Area of triangle is 78.893 cm².

Answer 2

Answer:

Given :-

• The perimeter of an isosceles triangle is 42 cm.
• The ratio of the equal side to its base is 3 : 4.

To Find :-

• What is the area of the triangle.

Formula Used :-

$\clubsuit$ Area Of Triangle by Heron's Formula :

$\footnotesize\mapsto \sf\boxed{\bold{\pink{Area_{(Triangle)} =\: \sqrt{s(s - a)(s - b)(s - c)}}}}$

where,

• s = Semi-Perimeter
• a = First Side Of Triangle
• b = Second Side Of Triangle
• c = Third Side Of Triangle

Solution :-

Let,

$\mapsto \bf First\: Side_{(Triangle)} =\: 3a\: cm$

$\mapsto \bf Second\: Side_{(Triangle)} =\: 4a\: cm$

$\mapsto \bf Third\: Side_{(Triangle)} =\: 3a\: cm$

As we know that :

$\footnotesize\bigstar\: \: \sf\boxed{\bold{\pink{Perimeter_{(Triangle)} =\: Sum\: of\: all\: sides}}}\: \: \bigstar$

Given :

• Perimeter of triangle = 42 cm

According to the question by using the formula we get,

$\implies \sf 42 =\: 3a + 4a + 3a$

$\implies \sf 42 =\: 10a$

$\implies \sf \dfrac{42}{10} =\: a$

$\implies \sf 4.2 =\: a$

$\implies \sf\bold{\purple{a =\: 4.2\: cm}}$

Hence, the required sides of a triangle are :

❒ First Side Of Triangle :

$\implies \sf First\: Side_{(Triangle)} =\: 3a\: cm$

$\implies \sf First\: Side_{(Triangle)} =\: (3 \times 4.2)\: cm$

$\implies \sf\bold{\green{First\: Side_{(Triangle)} =\: 12.6\: cm}}$

❒ Second Side Of Triangle :

$\implies \sf Second\: Side_{(Triangle)} =\: 4a\: cm$

$\implies \sf Second\: Side_{(Triangle)} =\: (4 \times 4.2)\: cm$

$\implies \sf\bold{\green{Second\: Side_{(Triangle)} =\: 16.8\: cm}}$

❒ Third Side Of Triangle :

$\implies \sf Third\: Side_{(Triangle)} =\: 3a\: cm$

$\implies \sf Third\: Side_{(Triangle)} =\: (3 \times 4.2)\: cm$

$\implies \sf\bold{\green{Third\: Side_{(Triangle)} =\: 12.6\: cm}}$

Now, we have to find the semi-perimeter of a triangle :

As we know that :

$\footnotesize\bigstar\: \: \sf\boxed{\bold{\pink{Semi-Perimeter =\: \dfrac{Sum\: of\: all\: sides}{2}}}}\: \: \bigstar$

Given :

• First Side (a) = 12.6 cm
• Second Side (b) = 16.8 cm
• Third Side (c) = 12.6 cm

According to the question by using the formula we get

$\implies \bf Semi-Perimeter =\: \dfrac{a + b + c}{2}$

$\implies \sf Semi-Perimeter =\: \dfrac{12.6 + 16.8 + 12.6}{2}$

$\implies \sf Semi-Perimeter =\: \dfrac{\cancel{42}}{\cancel{2}}$

$\implies \sf\bold{\purple{Semi-Perimeter =\: 21\: cm}}$

Now, we have to find the area of the triangle by using the Heron's Formula :

Given :

• Semi-Perimeter (s) = 21 cm
• First Side (a) = 12.6 cm
• Second Side (b) = 16.8 cm
• Third Side (c) = 12.6 cm

According to the question by using the formula we get,

$\small\longrightarrow \sf Area_{(Triangle)} =\: \sqrt{21(21 - 12.6)(21 - 16.8)(21 - 12.6)}$

$\small\longrightarrow \sf Area_{(Triangle)} =\: \sqrt{21(8.4)(4.2)(8.4)}$

$\small\longrightarrow \sf Area_{(Triangle)} =\: \sqrt{21 \times 8.4 \times 4.2 \times 8.4}$

$\small\longrightarrow \sf Area_{(Triangle)} =\: \sqrt{6223.392}$

$\small\longrightarrow \sf\bold{\red{Area_{(Triangle)} =\: 78.898\: cm^2}}$

${\small{\bold{\underline{\therefore\: The\: area\: of\: the\: triangle\: is\: 78.898\: cm^2\: .}}}}$