Maximum Angle

Geometry Level 2

In triangle $ABC$ , the three sides are $\overline{AB}=2$ , $\overline{BC}=4$ and $\overline{CA}=x.$ When angle $C$ is at its maximum, what is the value of $x^2?$

13 14 12 15

2 solutions

Kumar Gupta
Apr 23, 2014

When angle C is at a maximum, then $\cos \angle C$ is at a minimum.

By cosine rule, we have $\cos \angle C = \frac{ x^2 + 4^2 - 2^2 } { 2 \times x \times 4 } = \frac{ x^2+12 } { 8x }$ .

By AM-GM (or calculus), we know that the minimum of this occurs when $x^2 = 12$ .

Ashutosh Sharma
Apr 27, 2014

We may consider point B at origin. Let BC lie on +ve x-axis. So A must lie on a circle with radius 2 centered at origin . When AC is tangent to this circle Angle C will be maximum. And angle A will be 90. AB=2 BC=4, Just apply Pythagoras theorm

Maximum Angle

2 solutions

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