In the previous section we completed the discussion on percentage. In this chapter, we will discuss about the various properties of triangles.
• Any one vertex of the triangle
• Midpoint of the opposite side of the vertex under consideration
The fig.9.1(a) shows an example:
C is the vertex under consideration. So AB is the opposite side. D is the midpoint of AB. So CD is a median. Two more medians can be drawn. These are shown in fig(b). Here, E is the midpoint of CB, and F is the midpoint of AC. So AE and BF are medians. All the three medians of another triangle PQR is shown in fig(c). We can see that, the three medians intersect at a point.
A solution to this, is to draw the triangle on a card board, and make a cut out of it. Now we can place this triangle on a table as shown in fig (b). In the fig., the green line indicates the horizontal top surface of the table. When we place it on the table, Vertices A and B will be at the same level. Now we can measure the height. The height of triangle ABC is the height of vertex C from the top surface of the table. But the top surface of the table is same as side AB. So it the height from the side AB. We can measure it using a scale or a tape.
But the measurement should be done carefully. The height of C from AB can be taken in many different ways. This is shown in fig(c). CE, CF, CD, CG etc., give the distance of C from AB. They will all give different distances. We want the shortest distance. The distance will be shortest when the line along which the distance is measured from C to AB is perpendicular to AB. The line CD is such a line. It is perpendicular to AB. In fact, there will be only one perpendicular from a point to a line. So CD is our required line. The distance from C to D is the height of the triangle ABC. It is called the altitude. An altitude is the perpendicular line drawn from a vertex to the opposite side.
So we need not make a cutout of a triangle, and place it on the table, to find it’s altitude. All we need to do is, draw a perpendicular from a vertex to the opposite side. Fig(d) shows the altitude drawn without placing our triangle ABC on the table. Fig 9.3 below shows some more examples:
In the above fig.9.3(b), the perpendicular from R will not fall within PQ. So PQ is extended along the same line for a suitable distance, as shown by the dashed line. Similarly, in fig.(c) also, ON is extended for a suitable distance.
In the next section we will discuss about Exterior angles.
Median of a triangle
A line joining the following two points is called a median of a triangle:• Any one vertex of the triangle
• Midpoint of the opposite side of the vertex under consideration
The fig.9.1(a) shows an example:
 |
| Fig.9.1 |
C is the vertex under consideration. So AB is the opposite side. D is the midpoint of AB. So CD is a median. Two more medians can be drawn. These are shown in fig(b). Here, E is the midpoint of CB, and F is the midpoint of AC. So AE and BF are medians. All the three medians of another triangle PQR is shown in fig(c). We can see that, the three medians intersect at a point.
Altitude of a triangle
Consider the triangle ABC shown in fig.9.2(a) below. If we are asked to tell the height of this triangle, it will not be easy. Because the three vertices A,B and C are at three different levels. |
| Fig.9.2 |
But the measurement should be done carefully. The height of C from AB can be taken in many different ways. This is shown in fig(c). CE, CF, CD, CG etc., give the distance of C from AB. They will all give different distances. We want the shortest distance. The distance will be shortest when the line along which the distance is measured from C to AB is perpendicular to AB. The line CD is such a line. It is perpendicular to AB. In fact, there will be only one perpendicular from a point to a line. So CD is our required line. The distance from C to D is the height of the triangle ABC. It is called the altitude. An altitude is the perpendicular line drawn from a vertex to the opposite side.
So we need not make a cutout of a triangle, and place it on the table, to find it’s altitude. All we need to do is, draw a perpendicular from a vertex to the opposite side. Fig(d) shows the altitude drawn without placing our triangle ABC on the table. Fig 9.3 below shows some more examples:
 |
| Fig.9.3 |
In the next section we will discuss about Exterior angles.
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