Determining Coordinates

In 2D coordinate geometry, each point corresponds to a pair of numbers $(a,b)$ such that the first number $a$ gives the $x$-coordinate and the second number $b$ gives the $y$-coordinate of the point.

We use these coordinates to determine the placement of the point, as shown below:

Given a point in the coordinate plane, how do we determine the $x$-coordinate and $y$-coordinate of the point? It may be useful to follow these steps to help solve the problem:

• draw a picture;
• label the picture;
• mark the point or other desired quantity the problem is asking for;
• draw any additional lines or points that may help you find the solution.

Given point $(a,b)$, let's consider what happens when we reflect the point in the following ways:

• Reflect the point about the $x$-axis: this gives the point $(a,-b)$ with the same $x$-coordinate and with $y$-coordinate multiplied by $-1$.

• Reflect the point about the $y$-axis: this gives the point $(-a,b)$ with the same $y$-coordinate and with $x$-coordinate multiplied by $-1$.

• Reflect the point about the origin $(0, 0)$: this gives the point $(-a,-b)$ with both $x$- and $y$-coordinates multiplied by $-1$.

• Reflect the point about the line $y=x$: this gives the point $(b,a)$, with $x$- and $y$-coordinates swapped.

• Reflect the point about the line $y=-x$: this gives the point $(-b,-a)$, with $x$- and $y$-coordinates swapped and multiplied by $-1$.

Let's illustrate these concepts of reflection in the following example:

Consider the point $A=(2, 5).$ What are the coordinates of the reflections of $A$ about

• the $x$-axis
• the $y$-axis
• the origin $(0, 0)$
• the line $y=x$
• the line $y=-x$

respectively?

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The reflections of a point $(2, 5)$ about

• the $x$-axis is $(2, -5)$
• the $y$-axis is $(-2, 5)$
• the origin $(0, 0)$ is $(-2, -5)$
• the line $y=x$ is $(5, 2)$
• the line $y=-x$ is $(-5, -2).$ $_\square$

A set of points undergoes a translation if each point is mapped from $(x,y)$ to $(x+a, y+b)$ for some fixed numbers $a$ and $b$. To determine the coordinates of any point after this translation, add $a$ to the $x$-coordinate and $b$ to the $y$-coordinate of the point.

A parallel translation $(x, y) \rightarrow (x-3, y+b)$ moves the point $(-2, 5)$ to the point $(a, 7).$ What is $a+b?$

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The parallel translation $(x, y) \rightarrow (x-3, y+b)$ moves the point $(-2, 5)$ to

$$(-2-3, 5+b).$$

Equating this with the point $(a, 7)$ given in the problem, we have

$$-2-3=a,\quad 5+b=7,$$

which implies $a+b=-5+2=-3.$ $_\square$

Suppose that we are given a geometric figure and would like to determine the coordinates of one or more of the points of the figure. In this case, the following additional steps are useful:

• remember geometric properties of the figure in the problem;
• draw a diagram for the problem that illustrates these properties.

Parallelogram $ABCD$ has vertices

$$A=(-2, a),\ B=(b, 0),\ C=(3, 1),\ D=(1, 3).$$

What is $a+b?$

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Since $ABCD$ is a parallelogram, it must be true that the midpoints of $\overline{AC}$ and $\overline{BD}$ are the same. In other words,

$$\left(\frac{-2+3}{2}, \frac{a+1}{2}\right)=\left(\frac{b+1}{2}, \frac{0+3}{2}\right).$$

This implies $-2+3=b+1$ or $b=0,$ and $a+1=0+3$ or $a=2.$

Therefore, $a+b=2+0=2.$ $_\square$

Parallelogram $ABCD$ has vertex $A=(0, 2).$ If the midpoints of sides $\overline{AB}$ and $\overline{BC}$ are $X=(-1, 0)$ and $Y=(2, -1),$ respectively, what are the coordinates of vertex $D?$

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Let $B=(a, b),$ then

$$X=\left(\frac{0+a}{2}, \frac{2+b}{2}\right)=(-1, 0) \implies a=-2, b=-2.$$

Similarly, let $C=(c, d),$ then

$$Y=\left(\frac{-2+c}{2}, \frac{-2+d}{2}\right)=(2, -1) \implies c=6, d=0.$$

Thus, we have $B=(-2, -2)$ and $C=(6, 0).$ Now, let $D=(e, f),$ then since the midpoints of of $\overline{AC}$ and $\overline{BD}$ are the same in parallelogram $ABCD,$

$$\left(\frac{0+6}{2}, \frac{2+0}{2}\right)=\left(\frac{-2+e}{2}, \frac{-2+f}{2}\right) \implies e=8, f=4.$$

Therefore, $D=(8, 4).$ $_\square$