Question

It is known that the equations of the ribs of a triangle x = 0, y = m₁x + c₁, and y = m₂x + c₂. Prove that the area of the triangle is equal to (c₁ - c₂) ² / [2 (m₁ - m₂)]!

Answer 1

$\displaystyle \text{the second cut point of the y line }\\y=y\\m_1x+c_1=m_2x+c_2\\(m_1-m_2)x=c_2-c_1\\x=\frac{c_2-c_1}{m_1-m_2}\\\\t=\left|x\right|\\t=\left|\frac{c_2-c_1}{m_1-m_2}\right|\\t=\frac{c_2-c_1}{m_1-m_2}\\\\a=y_2-y_1\\a=c_2-c_1\\\\L=\frac12at\\L=\frac12(c_2-c_1)\frac{c_2-c_1}{m_1-m_2}\\L=\frac{(c_2-c_1)^2}{2(m_1-m_2)}\\\boxed{\boxed{L=\frac{(c_1-c_2)^2}{2(m_1-m_2)}}}$