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 \$

Other useful links

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 \$

Other useful links

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