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Let z(1),z(2) and z(3) be the vertices o...

Let `z_(1),z_(2) and z_(3)` be the vertices of trinagle . Then match following lists.

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The correct Answer is:
`a to s; b to p; c to q; d to r;`
a. `(z_(3) -z_(1))/(z_(2)-z_(1)) = (1+isqrt(3))/(2)`
`rArr (z_(3) -z_(1))/(z_(2) -z_(1)) = e^(i(pi)/(3))`
`rArr |(z_(3)-z_(1))/(z_(2)-z_(1)) = 1 and arg((z_(3)-z_(1))/(z_(2) -z_(2))) = (pi)/(3)`
So, triangle is equilateral.
b. `Re((z_(3)-z_(1))/(z_(3)-z_(2)))=0`
`rArr (z_(3)-z_(1))/(z_(3)-z_(2))` is purely imaginary
`rArr arg((z_(3)-z_(1))/(z_(1) -z_(2))) = pm (pi)/(2)`
So, triangle is right angled.
c. Let `(z_(3)-z_(1))/(z_(3) -z_(2)) = r (cos theta + i sin theta)`
Since `cos theta lt 0, arg((z_(3)-z_(1))/(z_(3)-z_(2))) gt (pi)/(2)`
So, triangle is obtuse angled.
d. `(z_(3) -z_(1))/(z_(3) -z_(2))=i`
`rArr |z_(3) -z_(1)| =|z_(3) -z_(2)|`
and `arg((z_(3)-z_(1))/(z_(3) -z_(2))) = (pi)/(2)`
So, triangle is isoceles right angled.
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