Coordinate geometry and describing movementIB MYP Maths Standard: Revision notes

Section 1

The Cartesian plane

The Cartesian plane has a horizontal $x$ -axis and a vertical $y$ -axis that cross at the origin $(0,0)$ . A point is written as coordinates $(x,y)$ : always across first, then up or down. The axes split the plane into four quadrants. Positive $x$ is to the right, positive $y$ is up. The point $(-3,4)$ is $3$ left and $4$ up, in the top-left quadrant. Plot points carefully, and check that the sign of each coordinate matches the direction.

Key termsCartesian planecoordinatesoriginquadrant

Common mistake

Writing coordinates as $(y,x)$ . Always write $x$ (across) first.

Section 2

Distances and shapes on a grid

When two points have the same $y$ -coordinate, the distance between them is the difference of the $x$ -coordinates (and the other way round). From $(-2,3)$ to $(4,3)$ is $4-(-2)=6$ . This lets you find the missing vertex of a rectangle with sides parallel to the axes: it takes the $x$ -coordinate of one neighbour and the $y$ -coordinate of the other. For rectangle $A(-2,3)$ , $B(4,3)$ , $C(4,-1)$ the fourth vertex is $D(-2,-1)$ and the area is $6\times4=24$ .

Exam tip

Subtract carefully when a coordinate is negative: $4-(-2)=6$ , not $2$ .

Section 3

Describing translations by movement

A translation moves every point of a shape by the same amount in the same direction. Describe it in words, for example ' $3$ right and $2$ up'. To find the movement, compare a point with its image: from $(-2,3)$ to $(4,-1)$ the $x$ -value goes up by $6$ and the $y$ -value goes down by $4$ , so the movement is $6$ right and $4$ down. To translate a point, add the right and up movements and subtract the left and down movements.

Key termstranslation

Common mistake

Mixing up the direction. A positive change in $x$ is right and a negative change in $y$ is down.

Section 4

Translating a shape

To translate a shape, move each vertex by the same movement and join them. For $P$ with vertices $(1,2)$ , $(4,2)$ and $(1,6)$ , a movement of $5$ left and $3$ down gives $(-4,-1)$ , $(-1,-1)$ and $(-4,3)$ . The image has the same side lengths and angles as the object.

Exam tip

Check one more vertex after you finish: if the movement works for all of them, the translation is right.

Section 5

Congruent shapes

Two shapes are congruent if they are exactly the same size and shape. One can be moved onto the other by a translation, reflection or rotation. On a grid, check that all the corresponding sides and angles are the same. A translated triangle is congruent to the original.

Key termscongruent

Section 6

Similar shapes

Two shapes are similar if one is an enlargement of the other: the corresponding angles are equal and the corresponding sides are all in the same ratio, called the scale factor. Triangle $T$ with legs $2$ and $3$ and triangle $U$ with legs $6$ and $9$ are similar, because $6\div2=3$ and $9\div3=3$ . A triangle with legs $4$ and $5$ is not similar to $T$ because $4\div2=2$ but $5\div3=1.67\ldots$ . All congruent shapes are similar (scale factor $1$ ), but similar shapes are only congruent when the scale factor is $1$ .

Key termssimilarscale factor

Common mistake

Checking only one pair of sides. Check every corresponding pair.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Coordinate geometry and describing movement

$ABCD$ is a rectangle with its sides parallel to the axes on a Cartesian plane. $A$ is $(-2,3)$ , $B$ is $(4,3)$ and $C$ is $(4,-1)$ . One unit on each axis is $1$ cm.
Describe the translation that maps $A$ onto $C$ , in terms of movement right or left and up or down.2 marks
Triangle $P$ has vertices $(1,2)$ , $(4,2)$ and $(1,6)$ on a Cartesian plane.
Triangle $Q$ has vertices $(6,2)$ , $(9,2)$ and $(6,6)$ . Describe the movement that maps $P$ onto $Q$ , and explain why $P$ and $Q$ are congruent.2 marks
Triangle $T$ has vertices $(0,0)$ , $(2,0)$ and $(0,3)$ . Triangle $U$ has vertices $(0,0)$ , $(6,0)$ and $(0,9)$ . Triangle $V$ has vertices $(0,0)$ , $(4,0)$ and $(0,5)$ .
Show that triangle $T$ and triangle $U$ are similar, and state the scale factor of the enlargement from $T$ to $U$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).