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$WXYZ$ is a square. If the co ordinates of $W$ and $Y$ are $(2,3)$ and $(5,7)$ respectively, find the co ordinates of $X$ and $Z$.

My Attempt:

I could only find the equation of diagonal $WY$ by using two point formula as follow $$(y-y_1)=\dfrac {y_2-y_1}{x_2-x_1} (x-x_1)$$ $$y-3=\dfrac {7-3}{5-2} (x-2)$$ $$4x-3y+1=0$$.

How do I complete the rest? please help

2 Answers 2

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First find the length of diagonal and it's slope. And also it's midpoint Then you can find the slope of other diagonal as diagonals are perpendicular and they bisect each other.

Then using slope ,midpoint find eq of second diagonal and use the length half the diagonal to find the two other vertices

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    Did you get it.? – 2017-02-24
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    No. Not clearly – 2017-02-24
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    Where is the problem please tell? – 2017-02-24
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    I couldn't understand your explanation in the second paragraph. – 2017-02-24
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    See you have found the slope of second diagonal the midpoint is common to that diagonal too as midpoint is where they intersect. So u can use slope point form of line to find its eq. Now length of first WY is 5 (we know W and Y. Si distance of each vertex from midpoint is 2.5. use this distance and the eq of second diagonal to find the other two vertices – 2017-02-24
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    @ ATHARVA, how do Use the distance and eqn to find the co ordinates. – 2017-02-24
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    Now take X you know distance between X and midpoint is 2.5. the co ordinates of midpoint is 3.5 ,5 . Now From the eq you express y coordinate of in terms of x and according to distance formula find distance in terms of x co ordinate an equate it with 2.5 .you will get a quadratic eq and thex co ordinate of both vertices – 2017-02-24
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    Did you get it now – 2017-02-24
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$$\begin{align} W(2,3)+(\;\;a,\;\;b)&=X(p,q)\qquad \cdots(1)\\ X(p,q)+(-b,\;\;a)&=Y(5,7)\qquad \cdots(2)\\ Y(5,7)+(-a,-b)&=Z(h,k)\qquad \cdots(3)\\ Z(h,k)+(\;\;b,-a)&=W(2,3)\;\;\;\quad \cdots(4)\end{align}$$ Adding $(1),(2)$ and solving gives $$(a,b)=\left(\frac 72, \frac 12\right)$$

Hence $$\color{red}{X=\left(\frac {11}2, \frac 72\right)\\ Z=\left(\frac 32, \frac {13}2\right)}$$ This satisfies $(3),(4)$ as well.

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    what is $(a,b)$ and how did you solve that? Please explain – 2017-02-26
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    $(a,b)$ is the change in $(x,y)$ coordinates from one vertex of the square to its neighbouring vertex. Draw a diagram and it will become clear. – 2017-02-26
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    @ hypergeometric, I couldn't draw. Could you please help? – 2017-02-26