the perimeter of triangle is 8 cm. if one of the sides is 3cm.what are the two sides for maximum area opf triangle?
Answers
Perimeter = 8 cm
Side = 3 cm
Find the length of the sides that will give maximum area:
To give the maximum area, the 2 other sides must be of equal length
⇒ The other two sides = (8 - 3) ÷ 2 = 2.5 cm
Find the area:
Semiperimeter = 8 ÷ 2 = 4m
Area = √p(p - a)(p - b)(p - c)
Area = √4(4 - 3)(4 - 2.5)(4 - 2.5)
Area = √9
Area = 3 cm²
Answer: Maximum area of the triangle = 3 cm²
If the perimeter of triangle is 8 then semi perimeter will be 4.
If length of the triangle is a, b, and c then let a be 3.
According to Heron’s formula, the area of triangle is given by;
A = root of [4(4 – 3) (4 – b) (4 – c)]
Since a + b + c = 8, b + c = 5, replacing it in above equation,
A = root of [4 * (4 – b) (4 – (5-b))]
Squaring on both sides,
A^2 = 4(4-b)(b-1)
On differentiating and setting it to 0
5 – 2b = 0
Second derivative will give negative value, hence,
B = 5/2, c = 5/2