### The perimeter of a triangle is 32 cm. One side of a triangle is 9 cm longer than the smallest side and the third side is 1 cm less than 4 times the smallest side. Find the area of the triangle.

**Answer:**

Area : 24 cm^{2}

**Step by Step Explanation:**

- Let's assume the smallest side of the triangle be
**x**cm. - According to the question, one side of the triangle is 9 cm longer than the smallest side.

The length of the side =**x**+ 9 - The third side is 1 cm less than 4 times the smallest side.

The length of the third side = 4**x**- 1 - The perimeter of the triangle is 32 cm.

Therefore,**x**+ (**x**+ 9) + (4**x**- 1) = 32

⇒**x**+**x**+ 9 + 4**x**- 1 = 32

⇒ 6**x**= 32 + 1 - 9

⇒**x**=24 6

⇒**x**= 4

Now,**x**+ 9 = 4 + 9 = 13,

4**x**- 1 = (4 × 4) - 1 = 15 - Therefore, all sides of the triangle are 4 cm, 13 cm and 15 cm.
- the following picture shows the triangle,

The area of the ΔABC can be calculated using Heron's formula, since all sides of the triangle are known.

S =

= 16 cm32 2

The area of the ΔABC = √S(S - AB)(S - BC)(S - CA)

= √16(16 - 4)(16 - 13)(16 - 15)

= 24 cm^{2}