Maximum Area...
Geometry
Level
2
In a triangle ABC, a = 3 and b = 2c. What is the maximum area of the triangle?
The answer is 3.
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1 solution
Solution 1:
Using Heron's formula the area A of the triangle (3,2c,c) is given by A² = (3c+3)(3-c)(3+c)(3c-3)/16 = 9(c²-1)(9-c²)/16
= 9/16 (-9+10c²-c^4) = 9/16 [16-(25-10c²+c^4)]
= 9/16 [16-(5-c²)²] = 9 - (5-c²)²/16
Can (5-c²)² = 0 ? This requires c² = 5 => c = √5 => b = 2√5.
b = 2c > c => a+b > c
9 > 5 => 3 > √5 => a > c => a+c > 2c = b
45 > 9 => 3√5 > 3 => 3c = b+c > a
So the triangle inequalities are verified. So c = √5 provides the max value for A².
So max value of A² = 9 => Max area A = 3