Question

The perimeter of an isosceles triangle is 44cm and the ratio of the equal side to its base is 4:3. The area of the triangle is *
12√55 cm²
4√55 cm²
3√55 cm²
16√55 cm²​

Answer 1

GiveN :

The perimeter of an isosceles triangle is 44cm and the ratio of the equal side to its base is 4:3.

To FinD :

The area of the triangle.

Solution :

Analysis :

Here the concept of Heron's Formula is used. We are provided with the ratio and the perimeter. First with the help of the perimeter we will find the three sides of the triangle and then by using Heron's Formula we can find the area of the triangle.

Required Formula :

• Semiperimter = Perimeter/2

• Heron's Formula = √[s(s-a)(s-b)(s-c)]

where,

• s is semiperimter
• a is first side
• b is second side
• c is third side

Explanation :

Let us assume that the three sides are "a" cm, "b" cm, "c" cm.

Now According to the question,

• Perimeter = 44 cm.

Let the common ratio be x.

Let the same side be a, b and the base be c.

• a = 4x cm
• b = 4x cm
• c = 3x cm

We know that all the sides of a triangle add upto its perimeter.

☯ According to the question,

⇒ 4x + 4x + 3x = 44

⇒ 11x = 44

⇒ x = 44/11

⇒ x = 4

∴ x = 4.

The sides are :

• a = 4x = 4 × 4 = 16 cm
• b = 4x = 4 × 4 = 16 cm
• c = 3x = 3 × 4 = 12 cm

Verification :

⇒ 16 + 16 + 12 = 44

⇒ 44 = 44

∴ LHS = RHS.

• Hence verified.

The area :

First we have to find out the perimeter.

We know that if we are provided with the perimeter of the triangle and is asked to find the semiperimter of the triangle then our required formula is,

Semiperimter = Perimeter/2

where,

• Perimeter = 44 cm

Substituting the values,

⇒ Semiperimter = Perimeter/2

⇒ Semiperimter = 44/2

⇒ Semiperimter = 22

∴ Semiperimter = 22 cm.

Now if we have the perimeter of the triangle and the three sides of the triangle and is asked to find the area of the triangle then our required formula is,

Heron's Formula = √[s(s-a)(s-b)(s-c)]

where,

• s is semiperimter = 22 cm
• a is first side = 16 cm
• b is second side = 16 cm
• c is third side = 12 cm

Using the required formula and substituting the required values,

⇒ Area = √[s(s-a)(s-b)(s-c)]

⇒ Area = √[22(22-16)(22-16)(22-12)]

⇒ Area = √[22 × 6 × 6 × 10]

⇒ Area = √[7920]

⇒ Area = √[2 × 2 × 2 × 2 × 3 × 3 × 55]

⇒ Area = 2 × 2 × 3√[55]

⇒ Area = 12√[55]

⇒ Area = 12√55

∴ Area = 12√55 cm².

Area of the isosceles triangle is 12√55 cm².