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Question 12 : The number of acute angled triangles whose sides are three consecutive positive integers and whose perimeter is at most 100 is

  1. 28
  2. 29
  3. 31
  4. 33

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Explanatory Answer

Perimeter is at most 100

Since three consecutive integers, Lets take random values

1, 2, 3
By triangle property sum of two sides has to be greater than third side
1 + 2 is not greater than 3
So, ruled out

Lets take 2, 3, 4
Acute angle property is Sum of square of any two sides has to be greater than square of third side

22 + 32 > 42

13 < 16. (So this condition is also not possible)

3, 4, 5 ----- Right angle triangle. (Not possible)

Lets take 4, 5, 6

42 + 52 > 62
41 > 36

This works (4,5,6)

Then (5,6,7) works

When the numbers become larger squares become increasingly larger
Numbers are close to each other, we are going closer to acute angle
So from (4,5,6),.......(32,33,34) Everything satisfies this condition.

32,33,34 -Perimeter is at most 100

So, out of 32 Combinations, First 3 are ruled out. So in total 29 combinations

The question is " The number of acute angled triangles whose sides are three consecutive positive integers and whose perimeter is at most 100 is "

Hence, the answer is 29

Choice B is the correct answer.

 

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