Geometry Practice : Right Triangle

This quant practice question is a problem solving question in geometry. Concept tested: Properties of inequalities satisfied by sides of right triangles.

Question

If ABC is a right angle triangle with angle A = 90^o and 2s = a + b + c, where a > b > c where notations have their usual meanings, then which one of the following is correct?

1. (s – b)(s – c) > s(s – a)
2. (s – a)(s – c) > s(s – b)
3. (s – a)(s – b) < s(s – c)
4. 4s(s – a)(s – b)(s – c) = bc

  Correct Answer      Choice (3). (s – a) (s – b) < s (s – c).

Explanatory Answer

Let a = 5, b = 4 & c = 3. Satisfies the condition that a > b > c.
These values form sides of a right angle triangle because 3, 4 and 5 is a Pythogorean triplet.
s = \\frac{a + b + c}{2}) = \\frac{12}{2}) = 6

Substitute these values and check which answer option holds good.
Choice 1: (s – b)(s – c) > s(s – a)
LHS: (6 - 4)(6 - 3) = 2 * 3 = 6
RHS: 6(6 - 5) = 6 * 1 = 6
LHS = RHS. LHS is not greater than RHS.
  Choice 1 is not the answer

Choice 2: (s – a)(s – c) > s(s – b)
LHS: (6 - 5)(6 - 3) = 1 * 3 = 3
RHS: 6(6 - 4) = 6 * 2 = 12
LHS < RHS. LHS is NOT greater than the RHS.
  Choice 2 is not the answer

Choice 3: (s – a)(s – b) < s(s – c)
LHS: (6 - 5)(6 - 4) = 1 * 4 = 4
RHS: 6(6 - 3) = 6 * 3 = 18
LHS < RHS as stated in the answer choice

  The correct answer is Choice (3).