Angles problem

Geometry Level 3

In a triangle $\Delta ABC$ , $2 \sin B\cos C= 1$ and the value of $\tan A$ is finite.

Which of its angles can be a right angle?

None of them

C

B

A

1 solution

Chew-Seong Cheong
Jan 31, 2017

Since $\tan A$ is finite; $\implies A \ne 90^\circ$ . Either $B$ or $C$ is a right angles. If $C=90^\circ$ , $2\sin B \cos 90^\circ = 0 \ne 1$ , $\implies C \ne 90^\circ$ . When $B=90^\circ$ , $2 \sin 90^\circ \cos C = 2 \cos C = 1$ , $\implies \cos C = \frac 12$ , $\implies C = 60^\circ$ and $A = 30^\circ$ ; and $\tan A = \tan 30^\circ$ is finite. Therefore $\boxed{B}$ is a right angle.

Though I got this correct; but I want to know whether all triangles satisfying these conditions will have angle B = 90 degrees? If no, then shouldnt the "none of these" option be considered too?

Yatin Khanna - 4 years, 4 months ago

You're right, not all triangles satisfying the given conditions will have $\angle B = 90^{\circ}$ , so I edited the question to "Which of the angles can be a right angle?", as only $\angle B$ can be a right angle.

Brian Charlesworth - 4 years, 4 months ago

Angles problem

1 solution

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