Question

Let N be the number of non-congruent triangles with all the following properties:
1. All three sides are integers less or equal to 15
2. The area of the triangle is an integer
3. At least one angle is 30, 45, 60, 120, 135, or 150
Compute N.
I require a solution that doesn't require a calculator.

Answer 1

Condition:

$\text{1. All 3 sides of the triangle are integers less or equal to 15}$

$\text{2. The area of the triangle is an integer}$

$\text{3. At least one angle is 30, 45, 60, 120, 145 or 150}$

To Find:

$\text{The number of such triangles.}$

Rule 1: All sides are integers.

$\text{According to the cosine rule : }a^2 = b^2 + c^2 - 2bc \cos A$

$\cos (30) = \dfrac{\sqrt{3} }{2} (\text {Rejected, this is an irrational number)}$

$\cos (45) = \dfrac{\sqrt{2} }{2} (\text {Rejected, this is an irrational number)}$

$\cos (60) =\dfrac{1}{2}$

$\cos (120) =-\dfrac{1}{2}$

$\cos (135) = -\dfrac{\sqrt{2} }{2} (\text {Rejected, this is an irrational number)}$

$\cos (150) = -\dfrac{\sqrt{3} }{2} (\text {Rejected, this is an irrational number)}$

$\textbf{So that left us with angle 60 and angle 120}$

Rule 2: The area is an integer

$\text{The area uses the following formula :} \dfrac{1}{2}ab\sin C$

$\sin (60) = \dfrac{\sqrt{3} }{2} (\text {Rejected, this is an irrational number)}$

$\sin (120) = \dfrac{\sqrt{3} }{2} (\text {Rejected, this is an irrational number)}$

$\textbf{So that left us with no possible angle in the list}$

$\boxed{ \boxed{\textbf{Answer: N = 0}}}$