### 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.

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**Answer: **192 cm^{2}

**Step by Step Explanation: **- 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, =

By cross multiplying, we get:

**b** = ------(1) - According to the question, the perimeter of the isosceles triangle ABC = 64 cm

Therefore, **x** + **x** + **b** = 64

⇒ 2**x** + = 64 **[From equation (1), ****b** = ]

⇒ = 64

⇒ 10**x** + 6**x** = 320

⇒ 16**x** = 320

⇒ **x** = 20 cm - Putting the value of
**x** in equation (1), we get:

**b** = = 24 cm - The area of the isosceles triangle ABC can be calculated using Heron's formula, since all sides of the triangle are known.

S = = 32 cm

The area of the isosceles triangle ABC = √S(S - AB)(S - BC)(S - CA)

= √32(32 - 24)(32 - 20)(32 - 20)

= 192 cm^{2} - Thus, the area of the triangle is 192 cm
^{2}.