Altitude

An altitude is a perpendicular dropped from one vertex to the side ( or its extension ) opposite to the vertex. It measures the distance between the vertex and the line which is the opposite side. Since every triangle has three vertices it has three altitudes .

1. Altitudes of an acute triangle :

   

   Figure 2.10

   For an acute triangle figure 2.10 all the altitudes are present in the triangle.

2. Altitudes for a right triangle :

   

   Figure 2.11

   For a right triangle two of the altitudes lie on the sides of the triangle, seg. AB is an altitude from A on to seg. BC and seg. CB is an altitude from C on to seg.AB. Both of them are on the sides of the triangle. The third altitude is seg. BD i.e.from B on to AC. The intersection point of seg. AB, seg. BC and seg. BD is B. Thus for a right triangle the three altitudes intersect at the vertex of the right angle.

3. Altitudes for an obtuse triangle :

   

   Figure 2.12

   D ABC is an obtuse triangle. Altitude from A meets line containing seg.BC at D. Therefore seg. AD is the altitude. Similarly seg.CE is altitude on to AB and BF is the altitude on to seg. AC. Of the three altitudes, only one is present inside the triangle. The other two are on the extensions of line containing the opposite side. These three altitudes meet at point P which is outside the triangle.

A line segment from the vertex of a triangle to the midpoint of the side opposite to it is called a median. Thus every triangle has three medians. Figure 2.13 shows medians for acute right and obtuse triangles.

Figure 2.13

All three medians always meet inside the triangle irrespective of the type of triangle.

A line segment from the vertex to the opposite side such that it bisects the angle at the vertex is called as angle bisector. Thus every triangle has three angle bisectors. Figure 2.14 shows angle bisectors for acute right and obtuse triangles.

Figure 2.14

Index

2.1 Introduction
2.2 Sum Of The Angles Of A Triangle
2.3 Types of Triangles
2.4 Altitude, Median And Angle Bisector
2.5 Congruence Of Triangles
2.6 Sides Opposite Congruent Angles

Chapter 3