How many triangles are there? Do it carefully.

39 42 40 41 44 43

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3 solutions

Chan Lye Lee
Dec 31, 2018

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 ( 5 2 ) × ( 3 1 ) = 30 {5 \choose 2} \times {3 \choose 1} =30 . Then consider those triangles containing NO vertex A A but vertex B B , there are ( 3 2 ) × ( 4 1 ) = 12 {3 \choose 2} \times {4 \choose 1} =12 such triangles. Hence there are a total of 42 \boxed{42} triangles.

A video for the solution.

i counted 53 ,

V i S i o N . - 2 years, 5 months ago

@V i S i o N . I think in your case, many triangles were double counted.

Chan Lye Lee - 2 years, 5 months ago

Nice!

Extension: What if there were 6 more lines emanating from the top vertex that intersect only the current intersections. Extension 2: Does the position of lines then change the answer?

Mahdi Raza - 8 months, 3 weeks ago
David Vreken
Dec 30, 2018

There are 42:

@David Vreken Thanks for your diagram.

Chan Lye Lee - 2 years, 5 months ago

Nice diagram! :)

Geoff Pilling - 2 years, 5 months ago
Jeremy Galvagni
Jan 3, 2019

Every triangle has a highest vertex. From each vertex, count how many triangles. Then add them up.

Interesting, could you please elaborate the approach?

Mahdi Raza - 8 months, 3 weeks ago

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