Coordinates: Mixed Practice

How to use this set#

Here are twelve ques­tions on the coor­di­nate plane. Please sketch every fig­ure, even a rough one. A quick sketch catches a sign error long before the arith­metic does. And when­ever you claim a shape, say which dis­tances you com­pared and why they set­tle the mat­ter.

Sec­tion A, Locat­ing points#

1. Write the quad­rant or axis of each point: $(-3, 7)$, $(0, -6)$, $(8, -8)$, $(-1, -1)$, $(5, 0)$.
2. A point $K$ has $y$-coor­di­nate $4$. Describe all the points like this. In which quad­rants can $K$ lie?
3. The points $A(-4, 0)$, $B(0, 3)$, $C(4, 0)$ and $D(0, -3)$ are joined in order. With­out plot­ting, say which pairs of points are mir­ror images in an axis, and name the axis each time. Then say what shape $ABCD$ is, and give your rea­son.
4. Plot $(3, 2)$, $(-3, 2)$, $(-3, -2)$ and $(3, -2)$. What shape do they form, and what are its side lengths?

Sec­tion B, Dis­tances#

5. Find the perime­ter of the tri­an­gle with ver­tices $(0, 0)$, $(8, 0)$ and $(8, 6)$.
6. Show that $(2, 3)$, $(8, 3)$ and $(5, 3 + 3\sqrt{3})$ are the ver­tices of an equi­lat­eral tri­an­gle.
7. Are the points $(-6, -2)$, $(0, 2)$ and $(9, 8)$ on one straight line?
8. A point $P$ is equidis­tant from $A(2, 0)$ and $B(0, 6)$ and lies on the $x$-axis. Find $P$.

Sec­tion C, Mid­points and cir­cles#

9. Find the mid­point of each side of the tri­an­gle $(2, 4)$, $(8, 4)$, $(6, 10)$.
10. The mid­points of the sides of a tri­an­gle are $D(4, 2)$, $E(7, 6)$ and $F(1, 5)$. Find the ver­tices. (Hint: the ver­tex oppo­site $D$ is found by using that $E$ and $F$ are mid­points of the two sides through it.)
11. Check whether $(6, 8)$, $(-10, 0)$ and $(0, -10)$ lie on a cir­cle cen­tred at the ori­gin. If so, find its radius, and decide whether $(7, 7)$ is inside, on or out­side it.
12. A screen is $640$ pix­els wide and $480$ high with the ori­gin at its bot­tom-left cor­ner. A cir­cu­lar icon of radius $60$ is cen­tred at $(70, 90)$; another of radius $90$ is cen­tred at $(200, 180)$. Does either icon run off the edge of the screen? And do the two icons over­lap?

Answers#