The Cartesian Plane: Quick Revision

A brief note on The Cartesian Plane. Read it before a test. Try each question at the end before you open its answer.

Coordinates

• A point is $(x, y)$: the abscissa $x$ goes along, and the ordinate $y$ goes up or down.
• The order matters: $(2, 5)$ and $(5, 2)$ are different points.
• The axes cross at the origin, $(0, 0)$.

Quadrants and axes

Reflections

In the x-axis $(x, y) \to (x, -y)$. In the y-axis $(x, y) \to (-x, y)$. In the origin $(x, y) \to (-x, -y)$.

Lengths and shapes

Along a line parallel to an axis, a length is the difference of the coordinate that changes. That gives the sides, the perimeter and the area of rectangles drawn on the grid.

A worked example

$A(-4, 3)$, $B(5, 3)$ and $C(5, -1)$ are three corners of the rectangle $ABCD$. Find $D$ and the area.

1. $D$. It shares $x$ with $A$ and $y$ with $C$: $D = (-4, -1)$.
2. The sides. $AB = 5 - (-4) = 9$ units and $BC = 3 - (-1) = 4$ units.
3. The area. $9 \times 4 = 36$ square units.

Practice with new numbers

A program to try

The program works out the worked example.

Mistakes to avoid

• Swapping the coordinates. The first number is always $x$.
• Putting a point on an axis into a quadrant. $(-3, 0)$ is on the x-axis.
• Changing the wrong sign in a reflection. The x-axis mirror changes $y$.
• Forgetting a minus sign in a length. From $-1$ to $3$ is $4$ units, not $2$.

Check yourself

1. In which quadrant is $(-1.5, -2.5)$?

2. What is the reflection of $(7, -3)$ in the x-axis?

3. How far apart are $(-4, 2)$ and $(-4, -6)$?

4. What are the coordinates of the point on the y-axis, $9$ units above the origin?