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Probability Level 3

Find number of right-angled triangles that can be made by joining vertices of a 1000 - sided regular polygon which has no side common with that of polygon .

Inspiration Aniket Sanghi

All of my problems are original


The answer is 497000.

Aryan Sanghi by Aryan Sanghi , 17, India

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1 solution

Aryan Sanghi
May 30, 2020

Now, let's see 3 observations

  1. All regular polygons are cyclic.

  2. Connecting 2-opposite vertices in a regular polygon with even number of vertices gives us a diameter of the circumcircle.

  3. Angle in a semicircle is 90°.

Now, there are 500 possible diameters and 2 of our vertices are ends of it. Now, we have to select the 3 r d 3^{rd} vertex which has 994 possible vertex(excluding the two vertices of our diameter and 4 vertices adjacent to those as we don't want a side common). So, there are 994 × 500 = 497000 \boxed{994 × 500 = 497000} possible triangles.

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Hey, why do people hit confuse button? You can ask your doubts in comments. It takes time and work to post questions and solutions.

Aryan Sanghi - 1 year ago

A little typo in your final answer as final answer is 497000 497000 .

Also, just saying that angle in a semicircle is 9 0 ∘ 90^{\circ} isn't sufficient I guess, you also need to show that in a circle one cannot make a right angled triangle without any of it's sides being the diameter of the circle.

Vilakshan Gupta - 1 year ago

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Isn't that understood that angle is 90° only in semicircle. Thanku for telling me of typo.

Aryan Sanghi - 1 year ago

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