The Cartesian plane

How do you tell a friend exactly where a seat is in a cin­ema? You give a row and a seat num­ber: two num­bers fix one place. Coor­di­nate geom­e­try does the same for points on a flat sur­face. Two per­pen­dic­u­lar num­ber lines turn every point into a pair of num­bers, and that lets us solve geom­e­try prob­lems with alge­bra. This les­son sets up the sys­tem, intro­duced by the French math­e­mati­cian René Descartes, which is why it is called the Carte­sian plane.

The axes and the ori­gin#

Draw a hor­i­zon­tal num­ber line and a ver­ti­cal num­ber line cross­ing at their zero points.

• The hor­i­zon­tal line is the x-axis. Num­bers to the right of zero are pos­i­tive, to the left neg­a­tive.
• The ver­ti­cal line is the y-axis. Num­bers above zero are pos­i­tive, below neg­a­tive.
• The cross­ing point is the ori­gin, $O$.

The two axes divide the plane into four regions called quad­rants, num­bered anti­clock­wise start­ing from the top right.

Coor­di­nates of a point#

To describe a point $P$, drop per­pen­dic­u­lars from $P$ to both axes. If the foot on the x-axis is at $x$ and the foot on the y-axis is at $y$, the coor­di­nates of $P$ are the ordered pair $(x, y)$.

• $x$ is the x-coor­di­nate or abscissa: the signed dis­tance of $P$ from the y-axis.
• $y$ is the y-coor­di­nate or ordi­nate: the signed dis­tance of $P$ from the x-axis.

The ori­gin is $(0, 0)$. The order mat­ters: $(2, 5)$ is $2$ to the right and $5$ up, while $(5, 2)$ is a dif­fer­ent point, $5$ to the right and $2$ up.

To plot $(-3, 2)$, start at the ori­gin, move $3$ units left, then $2$ units up, and mark the point.

Signs in the four quad­rants#

Points on the axes belong to no quad­rant. Every point on the x-axis has y-coor­di­nate $0$, so it looks like $(a, 0)$. Every point on the y-axis has x-coor­di­nate $0$, so it looks like $(0, b)$.

Exam­ple 1#

Where do these points lie? $(3, -4)$, $(-2, 5)$, $(-1, -7)$, $(0, 6)$, $(4, 0)$

$(3, -4)$ is in quad­rant IV, $(-2, 5)$ in quad­rant II, $(-1, -7)$ in quad­rant III. $(0, 6)$ lies on the y-axis and $(4, 0)$ on the x-axis.

Exam­ple 2#

How far is the point $(-6, 8)$ from each axis?

Its dis­tance from the x-axis is the size of its y-coor­di­nate, $8$ units. Its dis­tance from the y-axis is the size of its x-coor­di­nate, $6$ units. (Dis­tances are never neg­a­tive, so we drop the minus sign.)

Reflec­tions in the axes#

Fold­ing the plane along an axis sends each point to its mir­ror image.

• Reflec­tion in the x-axis: $(x, y) \to (x, -y)$.
• Reflec­tion in the y-axis: $(x, y) \to (-x, y)$.
• Reflec­tion in the ori­gin (through $O$ to the other side): $(x, y) \to (-x, -y)$.

So the reflec­tion of $(2, -5)$ in the x-axis is $(2, 5)$, in the y-axis is $(-2, -5)$, and in the ori­gin is $(-2, 5)$.

Shapes on the plane#

When the sides of a fig­ure run along grid lines, lengths can be read straight off the coor­di­nates.

Exam­ple 3#

A rec­tan­gle has ver­tices $A(1, 1)$, $B(5, 1)$, $C(5, 4)$ and $D(1, 4)$. Find its length, breadth and area.

$A$ and $B$ have the same y-coor­di­nate, so $AB$ is hor­i­zon­tal and its length is the dif­fer­ence of the x-coor­di­nates: $5 - 1 = 4$ units. $B$ and $C$ have the same x-coor­di­nate, so $BC$ is ver­ti­cal with length $4 - 1 = 3$ units. The area is $4 \times 3 = 12$ square units.

Exam­ple 4#

Find the area of the tri­an­gle with ver­tices $O(0, 0)$, $A(6, 0)$ and $B(0, 4)$.

$OA$ lies along the x-axis with length $6$ and $OB$ along the y-axis with length $4$. The axes are per­pen­dic­u­lar, so the tri­an­gle is right-angled at $O$ and its area is $\displaystyle \frac{1}{2} \times 6 \times 4 = 12$ square units.

Prac­tice#

1. Name the quad­rant or axis on which each point lies: (a) $(-4, 3)$ (b) $(5, -2)$ (c) $(-3, -3)$ (d) $(0, -7)$ (e) $(8, 0)$
2. Write the abscissa and ordi­nate of $(-9, 4)$.
3. Write the coor­di­nates of the point on the x-axis that is $5$ units to the left of the ori­gin.
4. Write the coor­di­nates of the point on the y-axis that is $3$ units below the ori­gin.
5. How far is the point $(7, -24)$ from the x-axis, and from the y-axis?
6. Write the reflec­tions of $(-3, 7)$ in the x-axis, in the y-axis and in the ori­gin.
7. The point $(a, b)$ lies in quad­rant II. In which quad­rant does $(b, a)$ lie?
8. A square has three ver­tices $(0, 0)$, $(5, 0)$ and $(5, 5)$. Find the fourth ver­tex and the area of the square.
9. Find the area of the tri­an­gle with ver­tices $(0, 0)$, $(8, 0)$ and $(0, -3)$.

Answers#