Geometric Counting - Triangle Spotlight

Probability Level 2

How many triangles can you find in this figure?

21 15 27 9

Zandra Vinegar by Zandra Vinegar

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4 solutions

Zandra Vinegar Staff
Aug 15, 2015

12 Helpful 1 Interesting 1 Brilliant 0 Confused

Brilliant work

Aakash Khandelwal - 5 years, 10 months ago

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:D Thanks!

Zandra Vinegar Staff - 5 years, 10 months ago

10 significant intersecting points and 3 sections vertically which is 30. But at least 1 triangle is the same on the other side so (10x3)-3=27

Joaquin Frago - 5 years, 10 months ago

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Proof of the equation ?

Vibhor Agarwal - 5 years, 10 months ago

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I can't find a relevant and understandable presentation.

Joaquin Frago - 5 years, 9 months ago
Lew Sterling Jr
Aug 19, 2015

C(3,1) • C(4,2) + C(3,1) • C(4,2) - 9 = 27

2 Helpful 2 Interesting 2 Brilliant 3 Confused
Eljay Rugas
Aug 18, 2015

It is like you will multiply the number of vertical sections (n) of each side by itself three times (because the triangle has 3 sides), denoted by the equation n^3, to get the number of triangles (x) from the figure.

Since the triangle has 3 vertical sections:

x = n^3

x = 3^3

x = 27

Also effective with 1, 2, 4, 5, ... vertical sections.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

I thought about it close to that way too -- hence the organization by 'vertical' stripes extending out. It reminded me of spotlights. :)

However, for me, there are 3 vertical beams coming out of each corner. These beams can each stand alone, or can be combined into groups of 2 or all 3. But anything that uses the lowest beam is going to also be counted by the symmetric perspective from the other corner.

CASE 1 So I pulled "uses the bottom edge" out as one case for which the left and right corners share the full set. Those are counted: only the lowest beam, the first two beams, all three beams, and within those sets, beam reach 1, beam reach 2, beam reach 3 ==> 9 altogether

CASE 2 & 3 - symmetrically counted for the 2 corners: - middle beam only x3 - top beam only x3 - middle and top beam x3

I should also note that the main reason I put so much time into counting this carefully was because my goal is to find a solution for a number of rays coming out from the bottom corners. :)

Zandra Vinegar Staff - 5 years, 10 months ago
Chris White
Aug 19, 2015

I just counted them and got 27... multiple choice really broke that since no numbers were bigger than 27 thus when i got there i simply stopped counting since at the point the answer was obvious... really no math involved here since you can count, any 6 year old should be able to count to 27.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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