Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angleIf AD bisects BAC, then BD = CD
The Angle Bisector Theorem states that the angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. In other words, if we have a triangle ABC and a line AD bisecting angle A, then the length of segment AB divided by the length of segment AC will be equal to the length of segment BD divided by the length of segment CD.
Mathematically, we can express this theorem as:
(AB/AC) = (BD/CD)
where AB and AC are the lengths of the two sides emanating from vertex A, and BD and CD are the lengths of the two segments of the opposite side cut by the angle bisector AD.
To prove this theorem, we can use the properties of similar triangles. If we draw a line parallel to BC passing through point D, we get two similar triangles ABD and ACD. By the definition of similarity, we have:
AB/BD = AC/CD
Multiplying both sides by BD * CD, we get:
AB * CD = AC * BD
Dividing both sides by AC * CD, we get:
AB/AC = BD/CD
Therefore, we have proven the Angle Bisector Theorem. This theorem has many applications in geometry, such as finding the length of a triangle’s side or finding the location of the incenter of a triangle.
More Answers:
How To Find The Centroid Of A Triangle: Step-By-Step Guide For BeginnersExplore The Properties And Formula Of The Median Of A Triangle: Its Applications
The Angle Bisector Theorem And Its Converse: Proving Angle Bisectors In Triangles