Mathematics

Mathematics

Problem

In which of the following situations, it is possible to have a ΔABC\Delta ABC ? (All symbols used have usual meaning in a triangle) a. (a+c−b)(a−c+b)=4bc(a + c - b)(a - c + b) = 4bc b. b2sin⁡2C+cos⁡22B=abb^2 \sin 2C + \cos^2 2B = ab c. a=3,b=5,c=7a = 3, b = 5, c = 7 and C=2π3C = \frac{2\pi}{3} d. cos⁡(A−C2)=cos⁡(A+C2)\cos\left(\frac{A-C}{2}\right) = \cos\left(\frac{A+C}{2}\right)
Intuitive explanation

A triangle exists only when its three sides satisfy the triangle inequality: the sum of any two sides must be strictly greater than the third side. Rather than checking all three inequalities separately each time, focus on whether the longest side is less than the sum of the other two—if that's true, the other two inequalities automatically hold. This single check is your gateway to quickly ruling out impossible triangles.

Intuitive explanation

A triangle exists only when its three sides satisfy the triangle inequality: the sum of any two sides must be strictly greater than the third side. Rather than checking all three inequalities separately each time, focus on whether the longest side is less than the sum of the other two—if that's true, the other two inequalities automatically hold. This single check is your gateway to quickly ruling out impossible triangles.

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