Question

Without using Pythagoras theorem, show that $A\left(4,4\right)$, $B\left(3,5\right)$ and $C\left(-1,-1\right)$are the vertices of a right angled triangle.

Answer 1

Step 1: Finding the slope of line $AB$:
The given points are $A\left(4,4\right)$ and $B\left(3,5\right)$
Slope of a line is given by:

$m=\frac{y_2-y_1}{x_2-x_1}$
Slope of line $AB$ $\left(m_1\right)=\frac{5-4}{3-4}$
$\Rightarrow m_1=-1$

Step 2: Finding the slope of line $AC$:
The given points are $A\left(4,4\right)$ and $C\left(-1,-1\right)$
Slope of a line is given by: $m=\frac{y_2-y_1}{x_2-x_1}$
Slope of line $AC$ $\left(m_2\right)=\frac{-1-4}{-1-4}$
$\Rightarrow m_2=1$

Step 3: Using the condition of perpendicular lines:

For proving a right angled triangle ,the sides $AB$ and $AC$ must be perpendicular which means that the multiplication of their slopes should be $-1$

$\Rightarrow m_1\times m_2=-1\times 1$
$\Rightarrow m_1\times m_2=-1$
$\Rightarrow$Lines $AB$ and $AC$ are perpendicular to each other.

Hence, $A\left(4,4\right)$, $B\left(3,5\right)$ and $C\left(-1,-1\right)$are the vertices of a right angled triangle.