Geometry – Coordinates and Transformations — Free IB MYP3 Mathematics Questions

1QuestionFinding Missing Coordinates in ShapesConcept Practice

5 marks~8 minCriterion A

Rectangle

ABCD

has three known vertices:

A(1, 2)

,

B(1, 5)

, and

C(6, 5)

. The diagonals of a rectangle bisect each other.

a

Calculate the midpoint of diagonal

AC

. [2]

b

Calculate the coordinates of vertex

D

, given that the midpoint of diagonal

BD

equals the midpoint of diagonal

AC

. [2]

c

A student claims that

D

must lie on the line

x = 1

because

A

has

x

-coordinate

1

. Explain why this claim is incorrect. [1]

Solutions

2QuestionFinding Missing Coordinates in ShapesConcept Practice

2 marks~3 minCriterion D

A city planner is designing a rectangular plaza with benches at each corner. On a coordinate grid where 1 unit = 1 metre, three corners are placed at

A(2, 1)

,

B(2, 6)

, and

C(7, 1)

.

Explain one reason why using a coordinate system to locate the missing corner is more accurate than estimating its position during construction. [2]

Solutions

3QuestionUnderstanding Midpoint Between Two PointsConcept Practice

5 marks~8 minCriterion C

A student claims that

C(4,\ 5)

is the midpoint of the line segment joining

A(1,\ 2)

and

B(7,\ 8)

.

a

Apply the midpoint formula to find the midpoint

M

of segment

AB

. [2]

b

Compare your answer to point

C

and explain whether the student's claim is correct. [2]

c

A second student says: "Any point with an

x

-coordinate of 4 must be the midpoint of

AB

." Justify whether this statement is correct or incorrect. [1]

Solutions

4QuestionTranslating Shapes on a GridConcept Practice

5 marks~8 minCriterion C

Triangle

ABC

has vertices

A(-2, 1)

,

B(0, 4)

, and

C(3, -1)

. A student translates the triangle by the vector

\begin{pmatrix} -3 \\ 2 \end{pmatrix}

and claims the image vertices are

A'(-5, 3)

,

B'(-3, 6)

, and

C'(0, 1)

.

a

Calculate the coordinates of

A'

,

B'

, and

C'

after the translation. [3]

b

Compare your calculated coordinates with the student's claimed image points and state whether the claim is correct. [1]

c

A second student translates only vertex

C(3, -1)

and records the image as

C'(0, -3)

. Explain what mistake this student likely made. [1]

Solutions

5QuestionCombining Multiple TransformationsConcept Practice

2 marks~3 minCriterion A

Triangle

ABC

has vertices at

A(1, 2)

,

B(3, 2)

, and

C(2, 5)

.

The triangle is first reflected in the

y

-axis, then translated 2 units to the left.

a

Apply the reflection in the

y

-axis to find the image of point

C

. [1]

b

Apply the translation to your answer from part (a) to find the final image of point

C

. [1]

Solutions

6QuestionComparing Original and Enlarged ShapesConcept Practice

2 marks~3 minCriterion A

Triangle

ABC

has vertices at

A(1,1)

,

B(3,1)

, and

C(2,4)

. It is enlarged by a scale factor of 2 about the origin to produce triangle

A'B'C'

, with vertices at

A'(2,2)

,

B'(6,2)

, and

C'(4,8)

.

a

Identify the vertex in triangle

A'B'C'

that corresponds to vertex

B

. [1]

b

Explain which side of triangle

A'B'C'

corresponds to side

BC

, and why. [1]

Solutions

7QuestionApplying Two or More Transformations SequentiallyConcept Practice

2 marks~3 minCriterion A

A triangle has vertices at

A(1, 2)

,

B(3, 2)

, and

C(2, 5)

.

The triangle is first reflected across the

y

-axis, then translated 3 units to the right.

Calculate the coordinates of vertex

C(2, 5)

after both transformations. [2]

Solutions

8QuestionPlotting Points in All Four QuadrantsConcept Practice

3 marks~5 minCriterion A

Three vertices of a rectangle are plotted on a Cartesian plane:

A(-2,\ 3)

,

B(4,\ 3)

, and

C(4,\ -1)

.

a

Calculate the coordinates of the fourth vertex,

D

. [1]

b

Identify the quadrant in which each of the four vertices

A

,

B

,

C

, and

D

is located. [1]

c

Explain how the signs of the

x

- and

y

-coordinates of a point determine which quadrant it lies in, using two of the four vertices as examples. [1]

Solutions

9QuestionFinding Missing Coordinates in ShapesAssessment Practice

4 marks~6 minCriterion B

Three vertices of a rectangle are plotted on a coordinate grid:

A(1,2)

,

B(5,2)

, and

C(5,6)

.

a

Plot points

A

,

B

, and

C

on the grid. Calculate the coordinates of the fourth vertex,

D

. [1]

b

A second rectangle has vertices

P(-3,1)

,

Q(4,1)

, and

R(4,5)

. Explain how you know where the fourth vertex

S

must lie, and state its coordinates. [1]

c

A third rectangle has three known vertices

X(2,3)

,

Y(7,3)

, and

Z(7,8)

. Analyse the coordinates of all five known vertices from parts (a) and (b) to write a general rule for finding the missing vertex of any axis-aligned rectangle. Apply your rule to find the missing vertex

W

. [2]

Solutions

10QuestionRotating Shapes Around the OriginAssessment Practice

6 marks~9 minCriterion B

Points

A(1,0)

,

B(0,1)

,

C(-1,0)

, and

D(0,-1)

are plotted on the coordinate grid provided.

a

Apply a 90° counterclockwise rotation about the origin to each point. Record the new coordinates and describe the pattern you observe. [2]

b

Explain how the pattern from part (a) predicts the image of

(3,-2)

after a 90° counterclockwise rotation, then calculate the image coordinates. [2]

c

A student claims that a 270° counterclockwise rotation about the origin maps

(3,-2)

to

(-2,-3)

. Analyse whether this claim is correct by applying the rotation rule twice from part (a) and justifying your conclusion. [2]

Solutions

11QuestionDescribing Transformations in WordsAssessment Practice

6 marks~9 minCriterion D

A city planner uses a coordinate grid to model a building's noon shadow. The building is triangle

ABC

with vertices

A(2, 1)

,

B(2, 5)

, and

C(5, 1)

. The model assumes the building is perfectly vertical, so the shadow is a reflection of triangle

ABC

across the x-axis.

a

Apply the reflection rule

(x, y) \rightarrow (x, -y)

to find the coordinates of

A'

,

B'

, and

C'

. [2]

b

Explain why a reflection across the x-axis correctly models the shadow of a perfectly vertical building. [2]

c

The building actually leans slightly to the right. Describe a transformation that would better model the real shadow, and explain why the reflection model is no longer sufficient. [2]

Solutions

12QuestionComparing Original and Enlarged ShapesAssessment Practice

6 marks~9 minCriterion D

A cartographer enlarges a master map using a scale factor of 2.5 to produce a wall map. The master map has a printing tolerance of ±0.3 cm per measurement. The distance between two cities on the master map is recorded as 12.0 cm.

a

Calculate the maximum possible error in the enlarged distance on the wall map. [2]

b

Explain how this error affects the usefulness of the wall map for navigation. [2]

c

Analyse why using a single scale factor to enlarge a map with measurement tolerances limits the map's reliability for precise navigation. [2]

Solutions

13QuestionDrawing Enlarged Shapes Using Grid MethodsAssessment Practice

5 marks~8 minCriterion B

Triangle

ABC

has vertices at

A(2, 1)

,

B(6, 1)

, and

C(6, 5)

. It is enlarged with centre of enlargement

(2, 1)

and scale factor

0.5

.

A student claims: "The enlarged image has one-quarter the area of the original triangle."

a

Calculate the coordinates of the vertices of the enlarged triangle

A'B'C'

. [1]

b

Calculate the area of triangle

ABC

and the area of triangle

A'B'C'

. Show all working. [2]

c

Justify whether the student's claim is correct, referring to the relationship between scale factor and area. [2]

Solutions

14QuestionComparing Original and Enlarged ShapesAssessment Practice

6 marks~9 minCriterion C

A student uses a photocopier to enlarge a rectangular photo. The original photo measures 5 cm by 8 cm. The photocopier is set to 150% enlargement. The printed copy measures 7.4 cm by 12.1 cm.

a

Calculate the scale factor for each dimension using the printed measurements. Express each answer as a decimal. [2]

b

The expected scale factor for a 150% enlargement is 1.5. Compare your scale factors from part (a) to this expected value and explain whether the photocopier produced an accurate enlargement. [2]

c

A student suggests that measuring just one dimension of the printed copy is enough to check the photocopier's accuracy. Analyse this suggestion, identifying two reasons why a single measurement may not be reliable. [2]

Solutions

15QuestionApplying Two or More Transformations SequentiallyAssessment Practice

6 marks~9 minCriterion B

A point

(x, y)

is transformed by the following sequence: reflect in the

y

-axis, then translate by the vector

\begin{pmatrix}4\\-1\end{pmatrix}

, then reflect in the

x

-axis.

a

Apply this transformation sequence to each point below and record the final coordinates. [2]

A(2,\ 3)

\quad

B(-1,\ 5)

\quad

C(0,\ -4)

\quad

D(3,\ -2)

b

Using your results from part (a), explain how you can write a single rule

(x', y') = (\ldots,\ \ldots)

that gives the final coordinates directly from

(x, y)

. State your rule. [2]

c

A student claims the rule gives the same result as applying the three steps one at a time. Choose a point not used in part (a), apply both methods, and justify whether the student's claim is correct. [2]

Solutions

16QuestionApplying Two or More Transformations SequentiallyAssessment Practice

7 marks~11 minCriterion C

A graphic designer places a logo at point

P(2,\ 1)

on a coordinate grid. She applies a translation of

(4,\ -3)

followed by a

90°

clockwise rotation about the origin. Her colleague claims the final position would be the same if the rotation were applied first.

a

Apply the translation

(4,\ -3)

then the

90°

clockwise rotation to

P(2,\ 1)

. State the final coordinates. [2]

b

Apply the

90°

clockwise rotation then the translation

(4,\ -3)

to

P(2,\ 1)

. State the final coordinates. [2]

c

Compare your results from parts (a) and (b) and analyse whether the colleague's claim is valid. In your answer, explain why the order of transformations matters and describe one consequence of assuming transformations can always be reordered. [3]

Solutions

17QuestionIdentifying Final Image After Multiple TransformationsAssessment Practice

6 marks~9 minCriterion D

A drone delivery service uses a coordinate grid to plan its flight path. The drone starts at point

A(2, 3)

. It translates by the vector

\begin{pmatrix}4\\-2\end{pmatrix}

to reach point

B

, then reflects over the line

y = -1

to reach the drop zone at point

C

.

a

Calculate the coordinates of point

B

. [2]

b

Calculate the coordinates of point

C

after the reflection over

y = -1

. [2]

c

Analyse one way in which using a 2D coordinate grid to model the drone's flight path may not reflect real-world conditions. Use the context to support your answer. [2]

Solutions

18QuestionUnderstanding Axes Symmetry in CoordinatesAssessment Practice

5 marks~8 minCriterion C

A student claims that quadrilateral

ABCD

, with vertices

A(2,3)

,

B(4,5)

,

C(6,3)

, and

D(4,1)

, is symmetric across the

y

-axis.

a

Apply the transformation rule

(x, y) \rightarrow (-x, y)

to calculate the reflected coordinates of

A

,

B

,

C

, and

D

. [2]

b

Explain why the reflected quadrilateral is or is not in the same position as the original. [1]

c

Justify whether the student's claim is true or false, using your results from part (a). [2]

Solutions

19QuestionDescribing Positions Using CoordinatesAssessment Practice

4 marks~6 minCriterion D

A city planner places three bus stops on a coordinate map at:

A(2, 3)

\quad

B(5, 3)

\quad

C(5, 7)

Each unit represents 100 metres. The actual road distances are:

A

to

B

= 350 m, and

B

to

C

= 500 m.

a

Calculate the straight-line distance, in metres, from

A

to

B

and from

B

to

C

. [2]

b

Explain why the coordinate model underestimates the actual walking distances, and suggest one way to improve the model. [2]

Solutions

20QuestionUnderstanding Axes Symmetry in CoordinatesAssessment Practice

5 marks~8 minCriterion B

Four points are plotted on the Cartesian plane:

P(3,\ 2)

,

\quad Q(-1,\ 4)

,

\quad R(-3,\ -2)

,

\quad S(1,\ -4)

a

Calculate the coordinates of the reflections of

P

,

Q

,

R

, and

S

across the

x

-axis, the

y

-axis, and the origin. [3]

b

Explain why reflecting any point

(a,\ b)

across the

x

-axis gives the image

(a,\ -b)

, referring to distance from the

x

-axis in your answer. [1]

c

Using your results from part (a), justify whether the reflection of a point across the

y

-axis and then across the

x

-axis gives the same image as a single reflection across the origin. [1]

Solutions

Geometry – Coordinates and Transformations — Free MYP3 Mathematics Practice Questions