Counting Triangles In The Crown

Probability Level 2

How many triangles exist in the figure above?

107 450 20 55

1 solution

Ashish Menon
Feb 5, 2016

First, let us make 1 triangle like the following one In this figure only $1$ triangle is there.

Now, let us make 2 triangles In this figure, there are a total of $5$ triangles.

Now, let us take 3 triangles In this figure there are a total of $14$ triangles

Now, let us take 4 triangles. In this figure there is a total of $30$ triangles

Now, if we observe carefully, the number of triangles in case 2 was $2^2$ more than that in case 1.
The number of triangles in case 3 was $3^2$ more that that in case 2.
The number of triangles in case 4 was $4^2$ more than that in case 3.
So, the number of triangles in case 5 (that is the question) should be $5^2$ than that of case 4.
So, the number of triangles in the following figure is 30(no. of triangles in case 4) + $5^2$
= $30+25$
= $55$
Hope, you like my solution :-)

Good approach.

Anish Harsha - 5 years, 4 months ago

Thank you...

Ashish Menon - 5 years, 4 months ago

Nice but you need to show that the last triangle is $5^2$ more. An alternate approach could be to fix the line at the bottom. There are 10 lines apart from that. Hence there are $\left( \begin{matrix} 10 \\ 2 \end{matrix} \right)$ triangles with that base. Other triangles can be counted manually.

Arulx Z - 5 years, 3 months ago

Yes , thanks nice to know that this question is being seen by people even now.

Ashish Menon - 5 years, 3 months ago

Counting Triangles In The Crown

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