Question

Point P is inside triangle ABC and equidistant from all three sides. If the measure of angle ABC=60°, what is the measure of angle PBC?

Answer 1

∠PBC = 30°

Explanation:

Given that <ABC = 60° and P is equidistance from AB, AC, BC

If P is equidistance from all the side then triangle ABC is equilateral

<ABC = <ACB = <CAB =60 and P is the center of the triangle (characteristics of an equilateral triangle)

If a perpendicular line is drawn from any of the angle through P to any of the opposite side it will divide the side into equal length

For instance AB = AC = BC = 2CM

If we draw a perpendicular line from <ABC through P to AC at point D

Then we will have a right angle triangle BCD : < BDC =90, BC = 2CM, CD =1CM

We can use sine rule to determine the angle

Sine BDC = sine CBD

BC CD

Sine 90 = sine CBD

2 1

1 = sine CBD

2 1

sin CBD = 0.5

<CBD = 30°

<CBD = <PBC = 30°