### The perimeter of an isosceles triangle is 64 cm. The ratio of the equal side to its base is 5 : 6. Find the area of the triangle.

192 cm2

Step by Step Explanation:
1. Let's assume, the lengths of the base and the equal sides of the isosceles triangle are b cm and x cm respectively.
Following figure shows the isosceles triangle ABC,

The ratio of the equal side to its base is 5 : 6.
Therefore,
 5 6
=
 x b

By cross multiplying, we get:
b =
 6x 5
------(1)
2. According to the question, the perimeter of the isosceles triangle ABC = 64 cm
Therefore, x + x + b = 64
⇒ 2x +
 6x 5
= 64  [From equation (1), b =
 6x 5
]

⇒
 10x + 6x 5
= 64
⇒ 10x + 6x = 320
⇒ 16x = 320
x = 20 cm
3. Putting the value of x in equation (1), we get:
b =
 120 5
= 24 cm
4. The area of the isosceles triangle ABC can be calculated using Heron's formula, since all sides of the triangle are known.
S =
 64 2
= 32 cm
The area of the isosceles triangle ABC = √S(S - AB)(S - BC)(S - CA)
= √32(32 - 24)(32 - 20)(32 - 20)
= 192 cm2
5. Thus, the area of the triangle is 192 cm2.