Geometry in 5 sec

Geometry Level 1

Let be X ( x 1 , x 2 ) , Y ( y 1 , y 2 ) X(x_{1}, x_{2}),\;Y (y_{1}, y_{2}) two points on the straight line a x + b y = c . ax+by = c.

Another point Z ( z 1 , z 2 ) Z(z_{1}, z_{2}) lies on the line a x + b y = d , ax+by = d, and its reflection with respect to the segment X Y ‾ \overline{XY} is the point Z 0 ( z 1 0 , z 2 0 ) Z_{0}(z^{0}_{1}, z^{0}_{2}) .

What relation holds between the coordinates of the point Z 0 ? Z_{0}?

a z 1 0 + b z 2 0 = ∣ d − c ∣ a\,z^{0}_{1}+b\,z^{0}_{2}=\mid d-c\mid c z 1 0 + d z 2 0 = ∣ a − b ∣ c\,z^{0}_{1}+d\,z^{0}_{2}=\mid a-b \mid b z 1 0 + a z 2 0 = ∣ d − c ∣ b\,z^{0}_{1}+a\,z^{0}_{2}=\mid d-c \mid c z 1 0 + b z 2 0 = ∣ a − c ∣ c\,z^{0}_{1}+b\,z^{0}_{2}=\mid a-c \mid a z 1 0 + d z 2 0 = ∣ d − c ∣ a\,z^{0}_{1}+d\,z^{0}_{2}=\mid d-c \mid

Alejandro Sainz by Alejandro Sainz , 36, Italy

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