Introduction to properties of angle bisectors:
Definition of an angle bisectors
An angle bisector is a line or a ray that passes though the vertex of an angle and that bisects the angle at the vertex.
The angle bisector of an angle is the locus of a point that is at the same perpendicular distance from the arms of the angle. T his property of the bisector of an angle is very important and is used in different application.
Description and Proof of the Property of an Angle Bisector
In the above diagram the angle at vertex O is formed bt the rays OA and OB, which are called the arms of the angle AOB. The ray OX bisects the angle OAB, that is angle AOX = angle BOX
Consider any point P on the angle dissector OX. Draw perpendiculars PQ on the ray OA and PR on the ray OB.
Considering triangles POR and POQ,
angle POR = angle POQ (definition of angle bisector)
angle PRO = angle PQO (both are right angles, as per construction)
OP = OP (common side for both the triangles)
Therefore, the triangles POR and POQ are congruent.
Hence, PQ = PR (corresponding parts of the congruent triangles)
Thus the perpendicular distances from any point on a angle bisector to the arms of the angle are congruent.
This property is used to construct the angle bisector of an angle with a compass and a ruler.
Please express your views of this topic online math tutor by commenting on blog.
Properties of Angle Bisectors of a Triangle
The above diagram shows the triangle ABC in which AX , By and CZ are the angle bisectors of angles A, B and C respectively.
An important property of angle bisectors in a triangle is, all the angle bisectors are congruent.
Let the point of intersection of all these bisectors be I.
Now, since I is a point which lies on all the three angle bisectors, the perpendicular distances on the three sides IP, IQ and IR are congruent.
Therefore, a circle can be drawn with I as center, touching all the three sides of the triangle. This is called the inscribed circle of the triangle and I is called the incenter of the triangle.
Definition of an angle bisectors
An angle bisector is a line or a ray that passes though the vertex of an angle and that bisects the angle at the vertex.
The angle bisector of an angle is the locus of a point that is at the same perpendicular distance from the arms of the angle. T his property of the bisector of an angle is very important and is used in different application.
Description and Proof of the Property of an Angle Bisector
In the above diagram the angle at vertex O is formed bt the rays OA and OB, which are called the arms of the angle AOB. The ray OX bisects the angle OAB, that is angle AOX = angle BOX
Consider any point P on the angle dissector OX. Draw perpendiculars PQ on the ray OA and PR on the ray OB.
Considering triangles POR and POQ,
angle POR = angle POQ (definition of angle bisector)
angle PRO = angle PQO (both are right angles, as per construction)
OP = OP (common side for both the triangles)
Therefore, the triangles POR and POQ are congruent.
Hence, PQ = PR (corresponding parts of the congruent triangles)
Thus the perpendicular distances from any point on a angle bisector to the arms of the angle are congruent.
This property is used to construct the angle bisector of an angle with a compass and a ruler.
Please express your views of this topic online math tutor by commenting on blog.
Properties of Angle Bisectors of a Triangle
The above diagram shows the triangle ABC in which AX , By and CZ are the angle bisectors of angles A, B and C respectively.
An important property of angle bisectors in a triangle is, all the angle bisectors are congruent.
Let the point of intersection of all these bisectors be I.
Now, since I is a point which lies on all the three angle bisectors, the perpendicular distances on the three sides IP, IQ and IR are congruent.
Therefore, a circle can be drawn with I as center, touching all the three sides of the triangle. This is called the inscribed circle of the triangle and I is called the incenter of the triangle.
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