Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 decade ago

# Only a genius can answer this....?

1. With any triangle, draw a line from one vertex to the other side, there are now 3 triangles in the diagram.

2. Add another line to the vertex, count the number of triangles.

3. Keep adding lines to the vertex until you have a set of data.

4. Predict the number of triangles if 10, 20,100 lines are drawn

5. Find the general rule for the number of triangles formed (T) if m lines are drawn.

Update:

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Update 2:

By saying draw a line from one vertex to the other side, I meant draw a line which seperates the triangle into two parts. There are 3 triangles because it is one large triangle and two smaller ones within it. Adding another line means to draw another line from the same vertex to the same side. Heres a hint, if you draw 2 lines from vertex, you will get 6 triangles....its a matter of counting correctly. The harder part is to find the pattern between the number of lines and the number of triangles.

Relevance
• smci
Lv 7

>1. With any triangle (ABC), draw a line from one vertex to the other side (P_1), there are now 3 triangles in the diagram.

No, there are only two! ABP_1 and ACP_1

Unless you meant "draw a line from each of the vertices ,B,C to the other side (P,Q,R)"

Is this what you meant? You need to clarify?!

In that case the number of resulting triangles *depends on whether the three lines all coincide at a single point*, i.e how you chose the lines (e.g. as the bisectors of angles would, they all meet at the incenter).

If they do coincide, you would get 6 triangles.

If they do not coincide, you would get 4 triangles + 3 quadrilaterals.

>2. Add another line to the vertex, count the number of triangles.

Depends entirely on how you choose the line, and whether they call came from the same vertex (e.g. A), or ABC, see above.

>3. Keep adding lines to the vertex until you have a set of data.

What on earth does "set of data" mean? Ya gotta define this.

You could have a "set of data" with the original triangle.

>4. Predict the number of triangles if 10, 20,100 lines are drawn

As per 1., it depends on how the lines were drawn, and from which vertex(/ices). We could set an upper and lower bound on T(n), the number of triangles after n lines have been drawn.

>5. Find the general rule for the number of triangles formed (T) if m lines are drawn.

If you supply the clarifications to above, I will have a go at the answer. Otherwise, it's way too vague.

• jim n
Lv 4

Draw you lines from the highest point on the triangle to the base. When we draw several such lines, they will divide the base into several segments.

Let say we draw 5 lines.

The base is divided into 6 segments.

So we have 6 triangles whose base is one segment wide.

But we also have triangles whose base is 2 segments wide. There are 5 of these.

And there are 4 triangles whose base is 3 segments wide,

3 triangles whose base is 4 segments wide,

2 triangles whose base is 5 segments wide,

and 1 triangle whose base is 6 segments wide.

All together, the total number of triangles is

T = 6 + 5 + 4 + 3 + 2 + 1.

Also

T = 1 + 2 + 3 + 4 + 5 + 6.

Adding these two equations, we get

2T = (6+1) + (5+2) + (4+3) + (3+4) + (2+5) + (1+6)

2T = 7 + 7 + 7 + 7 + 7 + 7 = 6*7 = 42

T = 21

It works the same for any number of line.

For example, with 10 lines, we get 11 segments,

2T = 11*12 = 132

T = 66.

With m lines, T = (1/2)(m+1)*(m+2).