If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
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Expert's answer
2011-11-10T08:23:27-0500
Proof: let the angle be ABC (B is the vertex with the angle). Let its angle bisector be BD. Assume a point E anywhere on the line BD. Join E to BC and BA PERPENDICULARLY because distance is always measured perpendicularly. Now we have two right triangles BAE and BCE. Now in these two triangles- Statements Reasons
Angle ABE = Angle CBE & & & & & & & & & & & & & & & & ( angle bisector) Side BE = Side BE & & & & & & & & & & & & & & & & & & & & & (common line to both the triangles) Angle BAE = Angle BCE & (right angles)(90*)
Hence by Angle-side-angle CONGRUENCE rule, triangle BAE is CONGRUENT TO triangle BCE. and by CPCT AE IS EQUAL TO CE. (which is the perpendicular distance).
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