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The Triangle and its Properties

Medians of Triangles

The students of class VII were being taken to a tour to Corbett National Park. They stayed there in a tent. The entrance in the tent was of a triangular shape as shown in the following figure.

Now can you tell what the height of the tent is?

As we can see in the above figure, the height of the tent is the length of the vertical pole which is standing in the centre of the tent.

Similarly, in any triangle, we can draw a perpendicular which represents its height. The perpendicular representing the height of a triangle is called the altitude of the triangle.

Look at the triangle PQR below.

Here, P is a vertex of ΔPQR and is the opposite side of the vertex P. is a perpendicular drawn from P to. Line segment is called the height or altitude of the triangle.

An altitude can be defined as follows.

An altitude of a triangle is the perpendicular drawn from a vertex to the opposite side of the triangle.”

Note: A triangle can have three altitudes.

In the above figure, , and are the three altitudes of ΔPQR.

"The point of intersection of the altitudes is called the orthocentre of the triangle."

Construction of Altitudes of a triangle
I. Using set-square

1. Draw a PQR.

2. From point R, draw a perpendicular on side PQ. Where the perpendicular meets the side PQ, name it as point X. XR is the altitude formed on side PQ.

3. In a similar way construct the altitude from point P to side QR and from side Q to line PR.
Thus, the altitudes obtained are XR, QZ and PY.


II. Using Compasses
1. Draw a PQR.
2. With P as centre, draw an arc on line QR cutting it at points A and B.
3. With A and B as centres, draw two intersecting arcs at points X and Y. Draw a line joining XY cutting the line QR at point M. Join PM.

4. With Q as centre draw an arc on side RP extended to cut it at points C and D. With C and D as centres, draw two intersecting arcs. Let this line intersect PR at point N. Join QN.

5. Similarly, draw altitude from point R on PQ cutting the line PQ at point L.
6. Join PM, RL and QN and name the meeting point of these three altitudes as O.

O is called the orthocentre of the PQR.

Remember

The altitudes of a triangle may not always lie inside it.

In an obtuse-angled triangle, the altitude drawn from the vertex of an acute angle lies outside the triangle. In this case, we have to extend the opposite side of the vertex from which the altitude is drawn.

For example,

In the above figure, ΔABC is an obtuse-angled triangle where ∠ABC is an obtuse angle. is the altitude of ΔABC drawn from the vertex A to extended side. Similarly, is the altitude drawn from the vertex C to extended side . And, is the altitude drawn from B to .

Now, observe the altitudes drawn in the triangles ΔPQR and ΔABC. 

It can be seen that altitudes in each triangle intersect each other at a common point. 

Thus, altitudes of a triangle are concurrent.

In  ΔPQR and ΔABC, G is the point of concurrence.

Let us look at another example now.

Example:

In triangle ABC, is perpendicular to such that =. Are the median and the altitude drawn from A to same?

Solution:

Here,

Therefore, is an altitude of ΔABC drawn from the vertex A to.

Also,

Therefore, is a median of ΔABC drawn from the vertex A to.

Thus, the altitude and the median drawn from A to are the same.

Let us consider the following triangle ABC.

In the given figure, A is the vertex of the triangle ABC and is the side opposite to vertex A. A line segment is drawn joining the point A and the point D, where D is the mid-point of.

Then, we say that is the median of ΔABC.

A median can be defined as follows.

The line segment joining any vertex of a triangle to the mid-point of its opposite side is called the median of the triangle.”

Now we know what a median is, can we tell how many medians can be drawn inside a triangle?

In a triangle, there are three vertices. Therefore, a triangle can have three medians, as shown in the following figure.

Here,, and are the three medians of ΔABC.

The medians of a triangle always lie inside the triangle.

From the figure, it can be observed that the medians , and intersect each other at a common point G. 

"The point of intersection of the medians is called the centroid of the triangle."

Thus, medians of a triangle are concurrent. 

The point where medians intersect each other is known as the point of concurrence.  

In the above given figure, G is the point of concurrence.

Now, in order to understand this concept in greater detail, let us take a look at the following video.



Construction of Median of Triangle
1. Draw a ABC.
2. With B and C as centres and radius more than half of BC, draw two arcs intersecting at points X and Y. Join XY thus meeting the line BC at point P.

3. With A and C as centres and radius more than half of AC, draw two arcs intersecting at points M and N. Join MN thus meeting the line AC at point Q.

4. Similarly, draw the perpendicular bisector of line AB meeting AB at point R.
5. Join AP, BQ and CR. Let the meeting point be O.

Point O is the centroid of ABC and AP, BQ and CR are the medians of sides BC, AC and AB respectively.

Now, let us look at an example.

Example 1:

In the triangle PQR, PS is a median and the length of = 6.5 cm. Find the length of.

Solution:

Here, PS is the median to the sideand we know that the median connects vertex to the midpoint of other side. Therefore, S is the mid-point of QR.

Therefore, = 2

= 2 × 6.5 cm

= 13 cm

Let us consider the following triangle ABC.

In the given figure, A is the vertex of the triangle ABC and is the side opposite to vertex A. A line segment is drawn joining the point A and the point D, where D is the mid-point of.

Then, we say that is the median of ΔABC.

A median can be defined as follows.

The line segment joining any vertex of a triangle to the mid-point of its opposite side is called the median of the triangle.”

Now we know what a median is, can we tell how many medians can be drawn inside a triangle?

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