Show that two sides of a triangle are perpendicular

Question

$A(1.2, 3.8)$ , $B(2, 1.8)$ , $C(5, 3)$ are points on a coordinate grid.

I need to know how to show that the line segments $\overline{AB}$ and $\overline{BC}$ are perpendicular. Not sure where to start on this one so just any help that would give a hint or help me get started would be appreciated.

All I know is that if the product of two gradients is -1 then the lines are perpendicular. Thanks.

Answer 1

There is $3$ nice answer on this question. But there is an elementary solution which you probably will like it. Use Pythagorean theorem!

$$|AB|^2=4+0.64$$ $$|BC|^2=9+1.44$$ $$|AC|^2=14.44+0.64$$

So $AB \perp BC$. That's it.

Answer 2

Slope of segment $\overline{AB}$ = $\dfrac{1.8-3.8}{2-1.2} = \dfrac{-2}{0.8} = \dfrac{-5}{2}$.
Slope of segment $\overline{BC}$ = $\dfrac{3-1.8}{5-2} = \dfrac{1.2}{3} = \dfrac{2}{5}$

The slope of segment $\overline{AB}$ is the negative reciprocal of the slope of segment $\overline{BC}$ and thus is perpendicular.

Answer 3

I am using vectors for this one.Since : $$\overrightarrow {a} \cdot \overrightarrow {b} = |\overrightarrow {a}||\overrightarrow {b}| \cos{\theta}$$Therefore, if two vectors are perpendicular $\implies \theta = 90^{\circ} \implies \cos \theta =0 \implies \overrightarrow {a} \cdot \overrightarrow {b} = 0 $
$\overrightarrow {AB} = -0.8 \hat i + 2\hat j$

$\overrightarrow {BC} = -3 \hat i + -1.2\hat j$

$\overrightarrow {AB} \cdot \overrightarrow {BC}= (0.8) \cdot (-3) +(2)(-1.2) = 0 \implies \overrightarrow {AB} \bot \overrightarrow {BC}$

Answer 4

The line segment $AB$ passes through points $A$ and $B$. So,the line $AB$ with equation (let) $y=m_1x+c_1$ satisfies $A(1.2,3.8)$ and $B(2,1.8)$.So $AB$ is satisfied by $$3.8=1.2m_1+c_1$$ and $$1.8=2m_1+c_1$$.Solving we get the equation of $AB$ with slope $m_1=-2.5$.

Similarly solving for slope $m_2$ for line $BC$ we get $m_2=0.4$.

So,$m_2\times m_1=-1$.Hence the lines are perpendicular.

If you didn't know this before you may note it as a theorem that if the product of slopes of two lines are $=-1$ then the lines are perpendicular to each other.