First slide

Geometry of complex numbers

Question

The points $A, B$ and $C$ represent the complex numbers $z_{1}, z_{2}, (1 - i) z_{1} + i z_{2}$ (where $i = \sqrt{- 1}$ ) respectively on the complex plane. The triangle $A B C$ is:

Moderate

A

Isosceles but not right angled
B

Right angled but not isosceles
C

Isosceles and right angled
D

None of the above

Solution

Since $A = z_{1}, B \equiv z_{2}, C \equiv (1 - i) z_{1} + i z_{2}$
$∴ A B = | z_{1} - z_{2} |, B C = | (1 - i) z_{1} + i z_{2} - z_{2} |$

$= | 1 - i | | z_{1} - z_{2} | = \sqrt{2} | z_{1} - z_{2} |$

and $C A = | (1 - i) z_{1} + i z_{2} - z_{1} |$

$= | i | | - z_{1} + z_{2} | = | z_{1} - z_{2} |$

$∴ A B = C A$ , and ${(A B)}^{2} + {(C A)}^{2} = {(B C)}^{2}$ .

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