Three points have coordinates $A(1,2)$ , $B(9,0)$ and $C(6,t)$ . Calculate the value(s) of t if : (a) $angleABC=90^@$ (b) AC is perpendicular to BC?

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Answer 1 · 2018-02-06T16:30:01

(a) $t=-12$ and (b) $t=5$ or $t=-3$

Slope of line joining $(x_1,y_1)$ and $(x_2,y_2)$ is $(y_2-y_1)/(x_2-x_1)$

Therefore slope of $AB$ is $(0-2)/(9-1)=-2/8=-1/4$ ,

slope of $BC$ is $(t-0)/(6-9)=-t/3$ and

slope of $AC$ is $(t-2)/(6-1)=(t-2)/5$ .

Further if two lines are perpendicular, product of their slopes is $-1$

Hence (a) if $/_ABC=90^@$ , we have $AB$ perpendicular to $BC$ i.e. $-1/4xx(-t/3)=-1$

or $t/12=-1$ i.e. $t=-12$

(b) if $AC$ is perpendicular to $BC$ , then

$(t-2)/5xx(-t/3)=-1$ i.e. $t(t-2)=15$

or $t^2-2t-15=0$ or $(t-5)(t+3)=0$ i.e. $t=5$ or $t=-3$

Three points have coordinates A(1,2)A(1,2), B(9,0)B(9,0) and C(6,t)C(6,t). Calculate the value(s) of t if : (a) angleABC=90^@angleABC=90^@ (b) AC is perpendicular to BC?