The sum of three sides of an isosceles triangle is 20 cm, and the ratio of an equal side to the base is 3 : 4, The altitude of the triangle is:

Given:

The sum of three sides of an isosceles triangle = 20 cm

Ratio of an equal side to the base = 3 : 4

Formula Used:

Pythagoras theorem: a² + b² = c²

Calculation:

Let the equal sides be 3x cm and the base be 4x cm.

Sum of the sides: 3x + 3x + 4x = 20

⇒ 10x = 20

⇒ x = 2

So, the equal sides are 3 × 2 = 6 cm and the base is 4 × 2 = 8 cm.

In an isosceles triangle, the altitude bisects the base.

So, half of the base = 8 / 2 = 4 cm.

Now, using the Pythagoras theorem in one of the right triangles:

Altitude² + (4 cm)² = (6 cm)²

⇒ Altitude² + 16 = 36

⇒ Altitude² = 20

⇒ Altitude = √20

⇒ Altitude = 2√5 cm

The altitude of the triangle is 2√5 cm.