Question

6. An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find
the area of the triangle.​

Answer 1

Answer:

The area of the isosceles triangle is 9 √15 cm².

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

We have given that,

Perimeter of isosceles triangle is 30 cm.

Equal sides of the isosceles triangle are 12 cm each.

We have to find the area of the triangle.

We know that,

Perimeter of isosceles triangle = Sum of all sides

⇒ 30 = s₁ + s₂ + s₃

⇒ 30 = 2 s₁ + s₃ - - [ ∵ s₁ = s₂ ]

⇒ 30 = 2 * 12 + s₃

⇒ 30 = 24 + s₃

⇒ s₃ = 30 - 24

⇒ s₃ = 6 cm

∴ The third side of the triangle is 6 cm.

Now, we know that,

Altitude drawn from the common vertex of equal sides to the third side in an isosceles triangle bisects the third side.

∴ CD = ½ * BC

⇒ CD = ½ * 6

⇒ CD = 3 cm

Now, in △ADC, m∠ADC = 90°

∴ ( AC )² = ( AD )² + ( CD )² - - [ Pythagors theorem ]

⇒ ( 12 )² = ( 3 )² + ( AD )²

⇒ 144 = 9 + AD²

⇒ AD² = 144 - 9

⇒ AD² = 135

⇒ AD = √135 - - [ Taking square roots ]

⇒ AD = √(15 × 9)

⇒ AD = 3 √15 cm

Now, we know that,

Area of triangle = ½ * Base * Height

⇒ A ( △ABC ) = ½ * BC * AD

⇒ A ( △ABC ) = ½ * 6 * 3 √15

⇒ A ( △ABC ) = 3 * 3 √15

⇒ A ( △ABC ) = 9 √15 cm²

∴ The area of the isosceles triangle is 9 √15 cm².

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Alternative Method:

Perimeter of triangle = 30 cm - - [ Given ]

∴ Semi perimeter of triangle = ½ * 30

∴ Semi perimeter of triangle ( s ) = 15 cm

The sides of the triangle are 12 cm, 12 cm and 6 cm.

• s₁ = 12 cm

• s₂ = 12 cm

• s₃ = 6 cm

• s = 15 cm

Now, by Heron's formula,

Area of triangle = √[ s * ( s - s₁ ) * ( s - s₂ ) * ( s - s₃ ) ]

⇒ Area of triangle = √[ 15 * ( 15 - 12 ) * ( 15 - 12 ) * ( 15 - 6 ) ]

⇒ Area of triangle = √[ 15 * 3 * 3 * 9 ]

⇒ Area of triangle = √[ 15 * 9 * 9 ]

⇒ Area of triangle = 9 √15 cm²

∴ The area of the isosceles triangle is 9 √15 cm².

Answer 2

Answer:

Perimeter of triangle = 30 cm - - [ Given ]

∴ Semi perimeter of triangle = ½ * 30

∴ Semi perimeter of triangle ( s ) = 15 cm

The sides of the triangle are 12 cm, 12 cm and 6 cm.

s₁ = 12 cm

s₂ = 12 cm

s₃ = 6 cm

s = 15 cm

Now, by Heron's formula,

Area of triangle = √[ s * ( s - s₁ ) * ( s - s₂ ) * ( s - s₃ ) ]

⇒ Area of triangle = √[ 15 * ( 15 - 12 ) * ( 15 - 12 ) * ( 15 - 6 ) ]

⇒ Area of triangle = √[ 15 * 3 * 3 * 9 ]

⇒ Area of triangle = √[ 15 * 9 * 9 ]

⇒ Area of triangle = 9 √15 cm²

∴ The area of the isosceles triangle is 9 √15 cm².

Step-by-step explanation: