How many triangle are there (part 2)?

How many triangles are there?

51 52 53 54 55 56 57

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

David Vreken
Jan 2, 2019

All possible triangles that can be formed with the blue quadrilateral below as part of its vertex must also contain 1 1 of the red segments below, for a total of 5 5 possible triangles:

Counting in this fashion, each of the 5 5 blue outer polygons have 5 5 possible triangles each, each of the 4 4 green polygons in the next layer have 4 4 possible triangles each, each of the 3 3 yellow polygons in the next layer have 3 3 possible triangles each, each of the 2 2 orange polygons in the next layer have 2 2 possible triangles each, and the last red triangle has 1 1 possible triangle.

This gives a total of 5 2 + 4 2 + 3 2 + 2 2 + 1 2 = 55 5^2 + 4^2 + 3^2 + 2^2 + 1^2 = \boxed{55} triangles.

Nice approach!

Chan Lye Lee - 2 years, 5 months ago
Chan Lye Lee
Jan 2, 2019

First note that any triangle obtained from this figure must

(i) be formed using 3 non-parallel lines, either 2 blue lines and 1 red line, OR 2 red lines and 1 blue line;

(ii) contain either the vertex A A or the vertex B B .

Let's consider those triangles containing the vertex A A . The number of triangles is ( 6 2 ) + ( 5 2 ) + ( 4 2 ) + ( 3 2 ) + ( 2 2 ) {6 \choose 2} +{5 \choose 2}+{4 \choose 2}+{3 \choose 2} +{2 \choose 2} . Then consider those triangles containing NO vertex A A but vertex B B , there are ( 5 2 ) + ( 4 2 ) + ( 3 2 ) + ( 2 2 ) {5 \choose 2} + {4 \choose 2} +{3 \choose 2} +{2 \choose 2} such triangles. Now note that ( n + 1 2 ) + ( n 2 ) = n 2 {{n+1} \choose 2} +{n \choose 2} = n^2 . Hence there are a total of 5 2 + 4 2 + 3 2 + 2 2 + 1 2 = 55 5^2+4^2+3^2+2^2+1^2= \boxed{55} triangles.

I made a video which is related to this problem, you may check it out.

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...