### 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.

Area : 24 cm2

Step by Step Explanation:
1. Let's assume the smallest side of the triangle be x cm.
2. According to the question, one side of the triangle is 9 cm longer than the smallest side.
The length of the side = x + 9
3. The third side is 1 cm less than 4 times the smallest side.
The length of the third side = 4x - 1
4. The perimeter of the triangle is 32 cm.
Therefore, x + (x + 9) + (4x - 1) = 32
x + x + 9 + 4x - 1 = 32
⇒ 6x = 32 + 1 - 9
x =
 24 6

x = 4
Now, x + 9 = 4 + 9 = 13,
4x - 1 = (4 × 4) - 1 = 15
5. Therefore, all sides of the triangle are 4 cm, 13 cm and 15 cm.
6. 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 =
 32 2
= 16 cm
The area of the ΔABC = √S(S - AB)(S - BC)(S - CA)
= √16(16 - 4)(16 - 13)(16 - 15)
= 24 cm2

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