Medians of a triangle

David Wells in one of his books wrote: if the triangle were a real physical sheet, made of some uniform material, it would not only have an area, but also a centre of gravity; see Fig. 1.3. This is the point on which the triangular sheet would balance on a pinpoint. We understand that, if the triangle is suspended from a vertex, a vertical line through that vertex will also pass through the centre of gravity. If the triangular sheet were resting on the edge of a table, it would start to tip and fall over the edge, if its centre of gravity were over the edge.

Fig. 1.3

You’ll find it natural to ask where the centre of gravity is. Actually, if the triangle is divided into numerous narrow parallel strips, each strip will balance about its midpoint, and all these midpoints appear to lie on the straight line joining the vertex to the midpoint of the opposite side (Fig. 1.4), called the median from that vertex.

Fig. 1.4

The centre of gravity of all the strips together will lie somewhere on the same straight line. Bearing in mind that we found area of the triangle in three different ways by starting with each side as base in turn, it is natural to do the same for the centre of gravity of the triangle. Three constructions can be made, each with a line of midpoints. If the centre of gravity lies on each of these lines, then there must be one point where all three lines meet. To find it, we’ll join each vertex to the midpoint of the opposite side, and the three lines concur, at the centre of gravity (Fig. 1.5). We have a bonus in this case, also. Because we are confident that the three medians do indeed concur. We do not even need to draw a diagram to check this fact, whereas we only discovered that the altitudes concurred with the aid of a drawing.

Fig. 1.5

Assignments:

1. Active vocabulary (memorize the following expressions and use them in sentences of your own):

a centre of gravity, to be suspended from, to bear in mind, to concur, to have a bonus, to be confident, with the aid of.

2. Answer the questions:

1. Where is the centre of gravity of a triangle?

2. What is a median of a triangle?

3. Do the three medians of a triangle necessarily concur?

4. What is the point where the medians concur called?

3. Agree or disagree:

1. If the triangle were a real physical sheet, made of some uniform material, it would not only have an area, but also a centre of gravity.

2. The median from a vertex is a straight line joining this vertex to the midpoint of the opposite side.

3. We can find area of the triangle in three different ways by starting with each side as base in turn.

4. Three medians of a triangle never concur.

4. Recall everything you know of Subjunctive Mood and insert suitable auxiliary verbs:

1. If the triangle were a real physical sheet, made of some uniform material, it … not only have an area, but also a centre of gravity.

2. If the triangular sheet were resting on the edge of a table, it … start to tip and fall over the edge.

3. If the triangle is suspended from a vertex, a vertical line through that vertex … also pass through the centre of gravity.

4. If the triangle … divided into numerous narrow parallel strips, each strip will balance about its midpoint.

5. If the centre of gravity lies on each of these lines, then there … one point where all three lines meet.

5. Complete the sentences, according to the text:

1. If the triangle were a real physical sheet …

2. If the triangle is suspended from a vertex …

3. If the triangular sheet were resting on the edge of a table …

4. If the triangle is divided into numerous narrow parallel strips …

5. If the centre of gravity lies on each of these lines …

6. If we were able to find area of the triangle in three different ways by starting with each side as base in turn …

7. If we were confident that the three medians did indeed concur …

6. Speak about medians of triangles and their properties, using expressions:

It is proved, obviously, evidently, apparently, there is no doubt, beyond question, indisputably, unquestionably.

Assignments for unit I:

1. Collect all the information about triangles, based on text “What is a triangle?”, according to your own plan.

2. Arrange the sentences, you have written out logically.

3. Unite all the sentences, using the following words and expressions: It is proved, obviously, evidently, apparently, there is no doubt, beyond question, indisputably, unquestionably, it is clear that…, considering, using hypotenuse as a side, to receive the square, is equal to, on the one hand, on the other hand, hence, it is evident that… It is known that…, we are quite familiar with…, every mathematician is sure of…, it should be pointed out…, in fact, thus, It is well known, consequently, therefore, so, it is obvious, it is evident, apparently, manifestly.

4. Do the same work (points 1, 2, 3) for each text: “Types of Triangles”, “Similarity criteria of triangles”, “Pythagorean Theorem”, “Medians of a triangle”.

5. Read the text, you have composed and make sure all the points of it are arranged logically.

6. Make changes if necessary.

7. Read the text one more time and think of the title.

Try to solve

Problem 1

Across the river

Jake Hardy was standing on the river bank, looking across to the far side. ‘How wide do you recon it is?’ asked Harold. Jake adjusted the rim of his hat, and turned to look downstream. He paused and then walked with deliberate paces along the river bank, then turned and called out, ‘About thirty meters, give or take a few.’

How did he estimate the width of the river?

Fig. 1.6