CBSE Class 9 Mathematics Loci and Concurrent Lines in Triangles Notes Set 08

Concurrent Lines and Special Points

Three or more lines passing through one point are concurrent. In a triangle, the angle bisectors meet at the incentre, the centre of the incircle that touches all three sides. The perpendicular bisectors of the sides meet at the circumcentre, which is equidistant from the vertices and is the centre of the circumcircle. The medians meet at the centroid, which divides each median in the ratio \( 2:1 \) measured from the vertex. The altitudes meet at the orthocentre.

Where the Points Lie

The circumcentre and orthocentre of an acute triangle lie inside it. In a right triangle the circumcentre is the midpoint of the hypotenuse and the orthocentre is the vertex of the right angle. In an obtuse triangle both lie outside. The centroid and incentre always lie inside. In an equilateral triangle all four points coincide; in an isosceles triangle they all lie on the axis of symmetry.

Locus

A locus is the set of all points that satisfy a given condition: every point on the locus satisfies the condition, and every point satisfying it is on the locus. The locus of points at distance \( r \) from a fixed point is a circle. The locus of points equidistant from two points is the perpendicular bisector of the segment joining them. The locus of points equidistant from the arms of an angle is its bisector. The locus of points equidistant from two parallel lines is the parallel line midway between them.

Using Loci in Problems

To show that points are collinear, show that they all satisfy the same locus condition: if \( PB=PC \), \( QB=QC \) and \( RB=RC \), then \( P,Q,R \) all lie on the perpendicular bisector of \( BC \). The centres of circles through two points \( A,B \) lie on the perpendicular bisector of \( AB \); the centres of circles touching a line at a point lie on the perpendicular to the line at that point; and the centres of circles touching two intersecting lines lie on the bisectors of the angles between them.

Worked Examples

Example 1. The medians of a triangle meet at G and the median AD is 12 cm long. Find AG.
G divides AD in the ratio \( 2:1 \), so \( AG=\dfrac23\times12=8 \) cm.

Example 2. Find the circumradius of a right triangle with legs 5 cm and 12 cm.
The hypotenuse is \( \sqrt{25+144}=13 \) cm and the circumcentre is its midpoint, so the circumradius is \( 6.5 \) cm.

Example 3. A point lies on the bisector of \( \angle XOY=80^\circ \) and is 4 cm from OX. What is its distance from OY?
Every point on the bisector is equidistant from both arms, so the distance is 4 cm.