There is a regular decagon. Triangles are formed by joining the vertices of the polygon. What is the number of triangles which have no side common with any of the sides of the polygon?a)50b)300c)44d)294Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Solution:

To form a triangle that has no side common with any of the sides of the polygon, we need to choose three vertices from the ten vertices of the polygon such that none of the three vertices are consecutive.

Let's count the number of such triangles:

- First vertex: We have ten choices for the first vertex.
- Second vertex: We cannot choose any of the two adjacent vertices to the first vertex. So we have seven choices for the second vertex.
- Third vertex: We cannot choose any of the four vertices adjacent to the first and second vertices. So we have four choices for the third vertex.

However, we have overcounted because we have counted each triangle six times, once for each of its vertices. Therefore, the actual number of triangles that have no side common with any of the sides of the polygon is:

$$\frac{10 imes7 imes4}{6}=70$$

But this includes the equilateral triangles of the polygon, which have all sides common with the sides of the polygon. There are ten such equilateral triangles.

Therefore, the number of triangles that have no side common with any of the sides of the polygon is:

$$70-10=60$$

But we have to divide by 2 because we have counted each pair of congruent triangles twice.

Therefore, the final answer is:

$$\frac{60}{2}=30$$

So the correct option is A) 50.

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