Coordinate Geometry Flashcards

All 18 cards in this deck

In a pair of coordinates $(x, y)$, what does the first number tell you?

e.g. $(3, -2)$

The $x$-coordinate — how far across (left or right) from the origin.

For $(3, -2)$: $3$ to the right.

What are the coordinates of the origin?

$(0, 0)$

True or false? The point $(4, -2)$ is in the same place as the point $(-2, 4)$.

False.

The order matters: $(4, -2)$ is $4$ across and $2$ down; $(-2, 4)$ is $2$ left and $4$ up.

A point lies below and to the left of the origin. What are the signs of its coordinates?

Both negative.

e.g. $(-3, -2)$

What are the steps to plot a point on a grid?

e.g. plot $(-4, 3)$

1. Start at the origin and move along the $x$-axis: $4$ to the left.
2. Then move parallel to the $y$-axis: $3$ up.
3. Mark with a cross and label it.

What is the midpoint of a line segment?

The point exactly halfway between the two endpoints.

What is the formula for the midpoint of the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$?

$\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$

What are the steps to find the midpoint from the coordinates of the two endpoints?

e.g. $A(2, 1)$ and $B(6, 7)$

1. Add the $x$-coordinates and halve: $(2+6)\div 2 = 4$.
2. Add the $y$-coordinates and halve: $(1+7)\div 2 = 4$.
3. Midpoint $= (4, 4)$.

True or false? The coordinates of a midpoint are always whole numbers.

False.

They can be decimals or fractions, e.g. the midpoint of $(-3, -2)$ and $(2, 4)$ is $(-0.5, 1)$.

True or false? The midpoint of $AB$ is the same point as the midpoint of $BA$.

True.

You can add the endpoint coordinates in either order before halving.

What does the gradient of a straight line measure?

Its steepness — how much $y$ changes for every $1$ that $x$ increases.

What is the formula for the gradient of the line through $(x_1, y_1)$ and $(x_2, y_2)$?

gradient $= \dfrac{y_2-y_1}{x_2-x_1}$ (change in $y$ $\div$ change in $x$)

What are the steps to work out the gradient of the line through two points?

e.g. $(1, 2)$ and $(4, 8)$

1. Change in $y$: $8 - 2 = 6$.
2. Change in $x$: $4 - 1 = 3$.
3. Divide: $6 \div 3 = 2$, so the gradient is $2$.

True or false? To find a gradient you divide the change in $x$ by the change in $y$.

False.

It is change in $y$ divided by change in $x$.

True or false? A line sloping downwards from left to right has a negative gradient.

True.

$y$ decreases as $x$ increases.

What are the steps to find a missing coordinate when you know the gradient and one point?

e.g. $A(5, 9)$, $B(d, 15)$, gradient of $AB$ is $3$

1. Put the values into the gradient formula: $\dfrac{15-9}{d-5} = 3$.
2. Rearrange: $6 = 3(d-5)$, so $d - 5 = 2$.
3. $d = 7$.

What are the steps to find the missing fourth vertex of a rectangle whose sides are parallel to the axes?
e.g. three vertices are $(1, 1)$, $(5, 1)$ and $(1, 4)$

1. Each $x$-coordinate should appear twice; the one appearing once is the missing $x$: from $1, 5, 1$ the missing $x = 5$.
2. Each $y$-coordinate should appear twice; the one appearing once is the missing $y$: from $1, 1, 4$ the missing $y = 4$.
3. Write the missing vertex as coordinates: $(5, 4)$.

True or false? A rectangle drawn on a coordinate grid always has its sides parallel to the axes.

False. A rectangle can be drawn tilted, so matching up repeated $x$- and $y$-coordinates only finds a missing vertex when the sides are parallel to the axes.