Coordinate Geometry Flashcards

All 25 cards in this deck

What does each number in the coordinate pair $(x, y)$ tell you?

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

First number = horizontal position from the origin (across), second = vertical position (up/down).

So $(3,-2)$ is 3 right and 2 down.

What are the coordinates of the origin?

$(0, 0)$

What are the signs of the coordinates of a point in the bottom-left quadrant?

Both negative.

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

What is true of the coordinates of every point on the $y$-axis?

The $x$-coordinate is $0$.

e.g. $(0, -4)$

True or false? $(4, -1)$ and $(-1, 4)$ are the same point.

False. Order matters: $(4,-1)$ is 4 right and 1 down, $(-1,4)$ is 1 left and 4 up.

True or false? A point with a negative $y$-coordinate lies below the $x$-axis.

True. Negative $y$ means downwards from the origin.

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 of a line segment from its endpoints?

e.g. $A(1, 3)$ and $B(7, 9)$

1. Add the $x$-coordinates and halve: $(1+7)\div 2 = 4$
2. Add the $y$-coordinates and halve: $(3+9)\div 2 = 6$
3. Write as a coordinate pair: $(4, 6)$

How do you find an endpoint when you know the midpoint and the other endpoint?

e.g. $A(2, 5)$, midpoint $M(6, 7)$, find $B$

Double each midpoint coordinate and subtract the known endpoint.

$B = (2\times 6 - 2,\ 2\times 7 - 5) = (10, 9)$

True or false? To find the midpoint you subtract the two $x$-coordinates and the two $y$-coordinates, then halve each.

False. You add each pair of coordinates and halve.

True or false? The midpoint of a line segment always has whole-number coordinates.

False. It can be a decimal or fraction.

e.g. midpoint of $(1,2)$ and $(2,4)$ is $(1.5, 3)$

What can you show about the diagonals of a quadrilateral on coordinate axes to prove it is a parallelogram?

Both diagonals have the same midpoint, so they bisect each other.

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

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$, i.e. the change in $y$ divided by the change in $x$.

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

e.g. $(1, 2)$ and $(4, 11)$

1. Change in $y$: $11 - 2 = 9$
2. Change in $x$: $4 - 1 = 3$
3. Divide: $9 \div 3 = 3$, so gradient $= 3$

What are the steps to find an unknown coordinate from a given gradient?

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

1. Put the coordinates into the gradient formula: $\dfrac{15-9}{d-5} = 3$
2. Multiply out: $6 = 3(d-5)$
3. Solve for the unknown: $d = 7$

What does a negative gradient tell you about a line?

It slopes downwards from left to right.

What is the gradient of a horizontal line?

$0$, because the change in $y$ is $0$.

True or false? A line of gradient $2$ is steeper than a line of gradient $-5$.

False. Steepness is judged by the size of the gradient ignoring the sign, so $-5$ is steeper.

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

$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ (Pythagoras' theorem)

What are the steps to find the length of a line segment from its endpoints?

e.g. $A(1, 2)$ and $B(4, 6)$

1. Differences: $4-1 = 3$ and $6-2 = 4$
2. Square and add: $3^2 + 4^2 = 25$
3. Square root: $\sqrt{25} = 5$

True or false? It does not matter which point you subtract from which when finding the length of a line segment.

True. The differences are squared, so the sign makes no difference.

True or false? The horizontal distance between two points with $x$-coordinates $-7$ and $8$ is $1$.

False. It is $8 - (-7) = 15$.

What can you work out to prove that a triangle on coordinate axes is isosceles?

The length of each side using Pythagoras, then show two lengths are equal.

Which coordinate calculation shows that two sides of a shape on coordinate axes are parallel?

Show the two sides have equal gradients.

e.g. gradient of $AB$ = gradient of $DC$ = $\frac{2}{3}$, so $AB$ is parallel to $DC$.

True or false? Showing that two sides of a quadrilateral on coordinate axes have equal length proves it is a rhombus.

False. You must show all four sides have equal length.