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Triangle Problem | PRMO-2018 | Problem No-24

Try this beautiful Trigonometry Problem based on Triangle from PRMO -2018, Problem 24.

Triangle Problem - PRMO 2018- Problem 24


If \mathrm{N} is the number of triangles of different shapes (i.e. not similar) whose angles are all integers (in degrees), what is \mathrm{N} / 100 ?

,

  • 15
  • 22
  • 27
  • 32
  • 37

Key Concepts


Trigonometry

Triangle

Integer

Suggested Book | Source | Answer


Suggested Reading

Pre College Mathematics

Source of the problem

Prmo-2018, Problem-24

Check the answer here, but try the problem first

27

Try with Hints


First Hint

Given that \mathrm{N} is the number of triangles of different shapes. Therefore the different shapes of triangle the angles will be change . at first we have to find out the posssible orders of the angles that the shape of the triangle will be different...

Now can you finish the problem?

Second Hint

case 1 : when x \geq 1 & y \geq 3 \geq 1

    \[x+y+z=180\]


={ }^{179} \mathrm{C}_{2}=15931
Case 2 : When two angles are same

    \[2 x+y=180\]


1,1,178
2,2,176
\vdots
89,89,2

Solution

But we have one case 60^{\circ}, 60^{\circ}, 60^{\circ}

    \[\text { Total }=89-1=88\]


Such type of triangle =3(88)
When 3 angles are same =1(60,60,60)
So all distinct angles's triangles

    \[\begin{array}{l}=15931-(3 \times 88)-1 \\neq 3 ! \=2611\end{array}\]


Now, distinct triangle =2611+88+1
=2700 \N=2700 \\frac{N}{100}=27 \

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Try this beautiful Trigonometry Problem based on Triangle from PRMO -2018, Problem 24.

Triangle Problem - PRMO 2018- Problem 24


If \mathrm{N} is the number of triangles of different shapes (i.e. not similar) whose angles are all integers (in degrees), what is \mathrm{N} / 100 ?

,

  • 15
  • 22
  • 27
  • 32
  • 37

Key Concepts


Trigonometry

Triangle

Integer

Suggested Book | Source | Answer


Suggested Reading

Pre College Mathematics

Source of the problem

Prmo-2018, Problem-24

Check the answer here, but try the problem first

27

Try with Hints


First Hint

Given that \mathrm{N} is the number of triangles of different shapes. Therefore the different shapes of triangle the angles will be change . at first we have to find out the posssible orders of the angles that the shape of the triangle will be different...

Now can you finish the problem?

Second Hint

case 1 : when x \geq 1 & y \geq 3 \geq 1

    \[x+y+z=180\]


={ }^{179} \mathrm{C}_{2}=15931
Case 2 : When two angles are same

    \[2 x+y=180\]


1,1,178
2,2,176
\vdots
89,89,2

Solution

But we have one case 60^{\circ}, 60^{\circ}, 60^{\circ}

    \[\text { Total }=89-1=88\]


Such type of triangle =3(88)
When 3 angles are same =1(60,60,60)
So all distinct angles's triangles

    \[\begin{array}{l}=15931-(3 \times 88)-1 \\neq 3 ! \=2611\end{array}\]


Now, distinct triangle =2611+88+1
=2700 \N=2700 \\frac{N}{100}=27 \

Subscribe to Cheenta at Youtube


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