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(A) Let $z = x + iy$. Then the points are $A(x, y)$,$B(x - y, x + y)$,and $C(0, x)$.Using the area formula for a triangle with vertices $(x_1, y_1), (

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$6$

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(A) Let $z = x + iy$. Then the points are $A(x, y)$,$B(x - y, x + y)$,and $C(0, x)$.Using the area formula for a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$:$	ext{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$	ext{Area} = \frac{1}{2} |x(x + y - x) + (x - y)(x - y) + 0(y - (x + y))|$$	ext{Area} = \frac{1}{2} |x(y) + (x - y)^2| = \frac{1}{2} |xy + x^2 - 2xy + y^2| = \frac{1}{2} |x^2 - xy + y^2|$Alternatively,using the property of complex numbers,the area of the triangle formed by $z_1, z_2, z_3$ is $\frac{1}{2} |	ext{Im}(\bar{z_1}z_2 + \bar{z_2}z_3 + \bar{z_3}z_1)|$.For $z_1 = z, z_2 = z + iz, z_3 = iz$:$	ext{Area} = \frac{1}{2} |	ext{Im}(\bar{z}(z + iz) + (\bar{z} - i\bar{z})(iz) + (-i\bar{z})(z))|$$	ext{Area} = \frac{1}{2} |	ext{Im}(|z|^2 + i|z|^2 + i\bar{z}z - i^2\bar{z}z - i|z|^2)| = \frac{1}{2} |	ext{Im}(|z|^2 + i|z|^2 + i|z|^2 + |z|^2 - i|z|^2)| = \frac{1}{2} |	ext{Im}(2|z|^2 + i|z|^2)| = \frac{1}{2} |z|^2$.Given $\frac{1}{2} |z|^2 = 18$,we get $|z|^2 = 36$,so $|z| = 6$.

If the area of the triangle formed by the points $z, z + iz$ and $iz$ on the complex plane is $18$,then the value of $|z|$ is

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