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 \

$