Counting Triangles

Number Theory Level 3

Count the number of triangles in the picture.

Don't try to actually count them!! ... think of another way to find the number!


The answer is 105.

Ossama Ismail by Ossama Ismail , 63, Egypt

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2 solutions

Hana Wehbi
Aug 3, 2017

Add the numbers above 1 + 2 + 3 + 4 + 5 + 6 = 21 1+2+3+4+5+6= 21

We have five lines ⟹ 5 × 21 = 105 \implies 5\times 21= 105 as an answer.

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Ossama Ismail
Aug 3, 2017

Each triangle consisting of two sides and a base. In the given figure there are 7 lines that you may select any two of them to be sides of a triangle, this makes the number of selections = 7 C 2 = ( 7 × 6 ) / 2 = 21 ^7C_2 = (7\times 6) / 2 =21 . To complete the triangle you need a base and in this case, there are 5 different bases.

then the total number of triangles = 7 C 2 × 5 = 21 × 5 = 105 ^7C_2 \times 5 = 21 \times 5 = 105

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