Question

Co-ordinates of the orthocentre of the trinagle whose vertices are $A(0,0),B(3,4)$ and $C(4,0)$ is

• A. $(3,1)$
• B. $(3,4)$
• C. $(3,3)$
• D. $(3,\frac{3}{4})$

Answer 1

The correct option is D $(3,\frac{3}{4})$

Triangle ABC, vertices are A(3,4), B(0,0), C(4,0)

O is the Orthocentre of the triangle

By considering the coordinates of B, C, A ,we can conclude that:

Equation of BC is y=0………..(1)

Equation of AD is x=3 ………..(2)

As we know slope of BC(being on the Xaxis) = 0

And for a vertical line AD, however the slope is not defined. It does not have a slope.

We take the slope of AC $=\frac{4–0}{3–4}=\frac{4}{-1}=-4$

So, the slope of its perpendicular(BE) has to be its negative reciprocal. That is the slope of BE $=\frac{1}{4}$

So, equation of BE, which is passing through (0,0) has to be y= mx + b , where

m = $\frac{1}{4}$, x=0, y=0

$\Rightarrow 0=\frac{1}{4}\times 0+b$

$\Rightarrow b=0$

So, BE is$y=\frac{1}{4}\times x+0$

$\Rightarrow y=\frac{x}{4}$

Now, by solving

$x=3$………..(1)

$y=\frac{x}{4}$ ………..(2)

We get $y=\frac{3}{4}$

So, Orthocentre coordinates are $(3,\frac{3}{4})$