Let a, b, c be the lengths of sides BC, CA, AB respectively of a triangle ABC. If p is the perimeter and q is the area of the triangle, then what is  equal to ?

Concept:

The perimeter of the triangle = sum of all sides

Area of triangle = (1/2)base × height

Calculation:

Since it is not mentioned, Δ ABC is which type of triangle. So, to make calculation easy

Let us assume, Δ ABC is a right-angle triangle where ∠A is 90°.

Let a = 5, b = 4, and c = 3 (Pythagorean triple)

Using the above concept, Perimeter

p = 3 + 4 + 5

⇒ p = 12  

Area of triangle

q = (1/2) 3 × 4

⇒ q = 6

Hence, the required value

 = 12(12 - 2 × 5)tan (90°/2) [∵ ∠A = 90°]

⇒ p(p − 2a)tan(A/2)= 24        [∵ tan 45° = 1

From option 4

4q = 4 × 6 = 24

∴ 4q is the correct answer.

Alternate MethodConcept:

The perimeter of the triangle = sum of all sides = a + b + c

Area of scalene triangle = , 

Half-angle formulae: 

Where 2s = a + b + c and a, b, c are the sides of the triangle.

Calculation:

If p be the perimeter of the triangle,

p = a + b + c                ------(1)

q =         ------(2)

where a, b, c are the sides of the triangle.

Now, 

 = (a + b+ c) (a + b+ c - 2a) 

⇒ (a + b+ c) (a + b+ c - 2a) 

⇒ (a + b+ c) (a + b+ c - 2a)