Perimeter of a triangle
The sum total length of all the three sides of a triangle is called its perimeter. In figure 4-8 the perimeter of the Triangle PQR
PQ QR PR 2cm 4cm 3cm 9cm
Median of a Triangle
The median of a triangle is the line segment joining a vertex to the mid-point of the opposite side. Consider a triangle PQR, mark the mid point of QR and name it S. Similarly mark the mid point of RP and PQ . Let them respectively be T and U. Now join P to S, Q to T and R to U. The lines PS, QT and RU are the medians of the triangle PQR. The three medians of a triangle are always concurrent at a point. Here, O is the point of concurrence of the median of Triangle PQR and is known as the Centroid of the triangle. Thus, Centroid is the point of concurrence of the medians of a triangle.
The centroid of a triangle divides the medians in a ratio 2:1 from the vertex of the triangle. In figure 4-9, OP=2 OS, OQ=2OT and OR=2OU. It should be noted that: a. Medians of an equilateral triangle are equal. b. Medians to the equal sides of an isosceles triangle are equal. c. The centroid of a triangle lies in the interior of the triangle.
Altitude of a Triangle
The altitude of a triangle is the line segment drawn from a vertex, perpendicular to the other side. In figure 4-10, PS QR from the vertex P to the side QR. Therefore, PS is the altitude of the triangle. Similarly, QT PR and RU PQ . Therefore, RU and QT are also the altitudes of PQR . The three altitudes are concurrent at O. O is called the Orthocentre of Triangle PQR . Thus, Orthocentre of a triangle is the point of concurrence of three altitudes of a triangle.
Every triangle has three altitudes. The altitudes of a triangle are concurrent.
Here G is the Orthocenter in each triangle. Notable points: 1. The orthocentre of an acute triangle lies in the interior of the triangle. 2. The orthocentre of a right triangle is the vertex of the right angle because sides PQ and QR can be considered as altitudes.
3. The orthocenter of an obtuse triangle lies in the exterior of the triangle. 4. Altitudes drawn on equal sides of an isosceles triangle are equal. 5. The altitude bisects the base of an isosceles triangle. 6. The altitudes of an equilateral triangle are equal.
Angle Bisectors of a Triangle
An angle bisector of a triangle is a line segment that bisects the angle of a triangle and its other end point lies on the side opposite to that angle. In figure 4-12 PL, QM and RN are the angle bisectors of the angles P, Q and R respectively. We observe that the angle bisectors PL, QM and RN are concurrent at O. This point O, is known as the incentre of the Triangle PQR . Therefore, Incentre of a triangle is the point of concurrence of the three angle bisectors of the triangle.
The incentre of a triangle lies in the interior of a triangle and is equidistant from the three sides of the triangle. Therefore, it is the centre of the incircle of a triangle.