Straight Line and Pair of Straight Lines 3 Question 11
12.
Using coordinate geometry, prove that the three altitudes of any triangle are concurrent.
(1998, 8M)
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Solution:
- Let the vertices of a triangle be, $O(0,0) A(a, 0)$ and $B(b, c)$ equation of altitude $B D$ is $x=b$.
Slope of $O B$ is $\frac{c}{b}$.
Slope of $A F$ is $-\frac{b}{c}$.
Now, the equation of altitude $A F$ is
$ y-0=-\frac{b}{c}(x-a) $
Suppose, $B D$ and $O E$ intersect at $P$.
Coordinates of $P$ are $[b, b \quad (\frac{(a-b)}{c})]$
Let $m _1$ be the slope of $O P=\frac{a-b}{c}$
and $\quad m _2$ be the slope of $A B=\frac{c}{b-a}$
Now, $\quad m _1 m _2=(\frac{a-b}{c}) \quad (\frac{c}{b-a})=-1$
We get, that the line through $O$ and $P$ is perpendicular to $A B$.
