Geometry – Properties of Shape — Free IB MYP3 Mathematics Questions

1QuestionProperties of Isosceles Equilateral and Right TrianglesConcept Practice

2 marks~3 minCriterion A

An isosceles triangle has two equal sides. One base angle measures

40°

.

Calculate the vertex angle of the triangle. [2]

Solutions

2QuestionCreating Symmetrical DesignsConcept Practice

2 marks~3 minCriterion A

A regular hexagon is shown in the diagram with one line of symmetry indicated by a dashed line.

a

State the total number of lines of symmetry of a regular hexagon. [1]

b

Explain how you know that a regular hexagon has two distinct types of lines of symmetry. [1]

Solutions

3QuestionTessellating Shapes and PatternsConcept Practice

2 marks~3 minCriterion B

The four regular polygons shown are an equilateral triangle, a square, a regular pentagon, and a regular hexagon.

a

Calculate the interior angle of each polygon using the formula

\dfrac{(n-2) \times 180°}{n}

, where

n

is the number of sides. [1]

b

Explain which of the four polygons can tessellate a plane, justifying your answer using the angles found in part (a). [1]

Solutions

4QuestionApplications of Congruence and Similarity in Real LifeConcept Practice

3 marks~5 minCriterion A

The diagram shows two rectangles. Rectangle A has dimensions 3 cm × 5 cm. Rectangle B has dimensions 6 cm × 10 cm.

a

Write the similarity statement for the two rectangles using the symbol

\sim

. [1]

b

Calculate the scale factor from Rectangle A to Rectangle B, expressing your answer as a fraction. [1]

c

A third rectangle, Rectangle C, is similar to Rectangle A with a scale factor of

\frac{3}{1}

from A to C. Determine the dimensions of Rectangle C and explain whether Rectangle B and Rectangle C could be the same rectangle. [1]

Solutions

5QuestionExploring Similar Shapes and ProportionalityConcept Practice

2 marks~3 minCriterion C

Rectangles

ABCD

and

PQRS

are similar. Rectangle

ABCD

has

AB = 3 \, \text{cm}

and

BC = 6 \, \text{cm}

. Rectangle

PQRS

has

PQ = 4 \, \text{cm}

and

QR = 8 \, \text{cm}

.

Describe the relationship between the corresponding sides of the two rectangles. Use appropriate mathematical notation and terminology in your response. [2]

Solutions

6QuestionDrawing Circles with Compass and RulerConcept Practice

2 marks~3 minCriterion A

The diagram shows a circle with centre

O

. Points

A

and

B

lie on the circumference, connected by a straight line segment. Point

C

also lies on the circumference, connected to

O

by a straight line segment.

a

Identify the centre, the radius, and the chord shown in the diagram. [1]

b

Explain why

OC

is a radius but

AB

is not. [1]

Solutions

7QuestionSolving Problems with Justification and ProofConcept Practice

2 marks~3 minCriterion A

The diagram shows two parallel lines cut by a transversal. Angle

a = 65°

is formed at the first parallel line. Angle

b

is formed at the second parallel line, in the same position relative to the transversal.

a

Identify the relationship between

\angle a

and

\angle b

. [1]

b

Explain why

\angle b = 65°

, using the relationship named in part (a). [1]

Solutions

8QuestionAlternate Interior and Exterior AnglesConcept Practice

2 marks~3 minCriterion B

The diagram shows two parallel lines cut by a transversal. Three pairs of alternate interior angles are formed:

- First pair:

30°

and

30°

- Second pair:

50°

and

50°

- Third pair:

70°

and

70°

a

Describe the pattern shown by the three pairs of angle measurements. [1]

b

Explain the general relationship between alternate interior angles when two parallel lines are cut by a transversal. [1]

Solutions

9QuestionComplementary Supplementary and Vertical AnglesConcept Practice

3 marks~5 minCriterion B

Four angle pairs are shown below.

Pair 1:

30°

and

60°

Pair 2:

45°

and

45°

Pair 3:

20°

and

70°

Pair 4:

10°

and

80°

a

Calculate the sum of each angle pair. [1]

b

Explain what these four pairs have in common, using the correct mathematical term for this type of angle relationship. [1]

c

A student claims that

35°

and

65°

are complementary angles. Analyse their claim and determine whether it is correct. [1]

Solutions

10QuestionAngle Relationships in Parallel Lines and TransversalsConcept Practice

2 marks~3 minCriterion A

In the diagram, two parallel lines are cut by a transversal. One angle at the first intersection is labeled

65^\circ

.

a

State the size of the corresponding angle at the second intersection. [1]

b

Explain why the two corresponding angles are equal. [1]

Solutions

11QuestionProperties of Isosceles Equilateral and Right TrianglesAssessment Practice

6 marks~9 minCriterion B

The table below shows the vertex angle and base angles of five isosceles triangles.

Vertex angle ( $v$ , degrees)	20	40	60	80	100
Base angle ( $b$ , degrees)	80	70	60	50	40

a

Calculate the sum of all three angles for each triangle in the table. [2]

b

Explain why the relationship between the vertex angle

v

and base angle

b

in any isosceles triangle can be written as

b = \dfrac{180 - v}{2}

. [2]

c

A student claims that an isosceles triangle with a vertex angle of

36°

has base angles of

72°

each, making it a special triangle where the base angle is exactly twice the vertex angle. Analyse whether this claim is correct and explain what is special about this triangle. [2]

Solutions

12QuestionProperties of Squares Rectangles Parallelograms and TrapeziumsAssessment Practice

5 marks~8 minCriterion C

Quadrilateral

ABCD

has vertices

A(1, 2)

,

B(4, 5)

,

C(7, 2)

,

D(4, -1)

.

a

Calculate the slopes of all four sides

AB

,

BC

,

CD

, and

AD

. [2]

b

Explain how your slopes show that

ABCD

is a parallelogram. [2]

c

Justify whether

ABCD

is a rectangle. Use co-interior angles in your answer. [1]

Solutions

13QuestionClassifying Triangles by Sides and AnglesAssessment Practice

6 marks~9 minCriterion D

A surveyor measures a triangular plot of land and records angles of

90°

,

45°

, and

45°

. Her laser tool has a margin of error of

\pm 2°

per angle.

a

Describe the type of triangle formed by the angles

90°

,

45°

, and

45°

, classifying it by both angles and sides. [2]

b

Explain how a

\pm 2°

measurement error could produce a different triangle. Give one possible set of actual angles and classify that triangle by both angles and sides. [2]

c

Analyse the validity of the surveyor's claim that the plot is exactly right isosceles, given the measurement error. [2]

Solutions

14QuestionExploring Regular and Irregular TessellationsAssessment Practice

6 marks~9 minCriterion C

A shape is formed by taking a regular hexagon with side length 2 cm and removing one corner equilateral triangle of side length 1 cm.

a

Calculate the interior angle at the concave (indented) vertex of the new shape. [2]

b

Explain why the angles at the concave vertex cannot combine with angles from other copies of the shape to sum to

360°

. [2]

c

Justify whether this shape can tessellate the plane. [2]

Solutions

15QuestionCreating Symmetrical DesignsAssessment Practice

6 marks~9 minCriterion D

A graphic designer proposes a logo for a wind turbine company using

6

-fold rotational symmetry, where the design repeats every

60°

. The manufacturer states it can only reproduce designs with a maximum of

4

-fold rotational symmetry, where the design repeats every

90°

.

a

Describe what

6

-fold rotational symmetry means for the appearance of the logo. [1]

b

Explain how the manufacturing limitation affects both the logo's appearance and the production process. [3]

c

The designer proposes switching to

3

-fold rotational symmetry (

120°

rotation) as a compromise. Analyse whether this compromise is suitable for the wind turbine company, identifying one advantage and one limitation. [2]

Solutions

16QuestionApplications of Congruence and Similarity in Real LifeAssessment Practice

5 marks~8 minCriterion B

Three pairs of similar rectangles are shown below.

Pair 1:

2 \text{ cm} \times 3 \text{ cm}

and

4 \text{ cm} \times 6 \text{ cm}

Pair 2:

3 \text{ cm} \times 4 \text{ cm}

and

6 \text{ cm} \times 8 \text{ cm}

Pair 3:

4 \text{ cm} \times 5 \text{ cm}

and

8 \text{ cm} \times 10 \text{ cm}

a

Calculate the scale factor from the smaller rectangle to the larger rectangle for each pair. [1]

b

Calculate the area of each rectangle in all three pairs, then calculate the ratio of the larger area to the smaller area for each pair. [2]

c

A new pair of similar rectangles has side lengths

5 \text{ cm} \times 7 \text{ cm}

and

15 \text{ cm} \times 21 \text{ cm}

. Using the relationship between the side scale factor and the area ratio you identified in part (b), justify the area ratio for this new pair without calculating the areas. [2]

Solutions

17QuestionApplications of Congruence and Similarity in Real LifeAssessment Practice

2 marks~3 minCriterion D

A surveyor wants to find the width of a river without crossing it. Standing on one bank, she measures a baseline of known length along the bank and records the angles from each end of the baseline to a landmark on the opposite bank. She then constructs a smaller triangle with the same angles.

Explain how the surveyor uses similar triangles to calculate the river's width, and state one assumption she must make for the result to be accurate. [2]

Solutions

18QuestionExploring Tangents and Secants IntroductoryAssessment Practice

7 marks~11 minCriterion C

In the diagram,

O

is the centre of a circle and line

PT

is a tangent to the circle at point

T

, with radius

OT

drawn to the point of tangency.

a

Describe what a tangent to a circle is. [1]

b

Explain why

OT

is the shortest distance from centre

O

to line

PT

. [2]

c

Justify why

OT

must be perpendicular to

PT

, using the diagram provided. [4]

Solutions

19QuestionExploring Tangents and Secants IntroductoryAssessment Practice

6 marks~9 minCriterion D

A satellite dish has a circular base with centre

O

and radius

1.5 \text{ m}

. A technician stands at point

P

, which is

4 \text{ m}

from

O

. Support cables run from

P

to points

A

and

B

on the edge of the dish, where

PA

and

PB

are tangents to the circle.

a

Calculate the length of cable

PA

. [2]

b

The technician measures

OP

as

4 \text{ m}

, but this measurement has an error of

\pm 5 \text{ cm}

. Calculate the range of possible cable lengths and explain what this means for the installation. [3]

c

The dish is slightly oval rather than circular. Explain one way this shape difference affects the reliability of the cable length calculated in part (a). [1]

Solutions

20QuestionExploring Tangents and Secants IntroductoryAssessment Practice

6 marks~9 minCriterion B

Point

P

lies outside a circle. Two tangent segments are drawn from

P

to the circle, touching it at points

A

and

B

.

The table below shows measurements from four different circles.

Circle 1	$PA = 5$ cm	$PB = 5$ cm
Circle 2	$PA = 7$ cm	$PB = 7$ cm
Circle 3	$PA = 9$ cm	$PB = 9$ cm
Circle 4	$PA = 12$ cm	$PB = 12$ cm

a

State a conjecture about the lengths of two tangent segments drawn from the same external point to a circle. [1]

b

A fifth circle has an external point

Q

, with tangent segments

QX = 8

cm and

QY = 8

cm. Explain whether this observation supports your conjecture. [2]

c

The diagram shows point

P

outside a circle with centre

O

. Tangent segments

PA

and

PB

touch the circle at

A

and

B

. Analyse the triangles

OAP

and

OBP

to explain why

PA

must equal

PB

. [3]

Solutions

21QuestionAlternate Interior and Exterior AnglesAssessment Practice

3 marks~5 minCriterion A

Two parallel lines are cut by a transversal. One alternate interior angle measures

65^\circ

and the other is labelled

x

.

a

Calculate the value of

x

. [1]

b

Explain why alternate interior angles are equal when two parallel lines are cut by a transversal. [1]

c

A student claims that if

x = 90^\circ

, both parallel lines would be perpendicular to the transversal. Do you agree? Justify your answer. [1]

Solutions

22QuestionAlternate Interior and Exterior AnglesAssessment Practice

4 marks~6 minCriterion D

A city planner is checking whether Main Street and Oak Avenue are parallel. Elm Street crosses both roads. A surveyor measures the alternate interior angle at Main Street as

118°

and the alternate interior angle at Oak Avenue as

116°

.

a

State the relationship between alternate interior angles when two lines are parallel. Are the two streets parallel? Justify your answer. [2]

b

Explain two real-world factors that could cause the

2°

difference in the measured angles, even if the streets were designed to be parallel. [2]

Solutions

23QuestionAngle Relationships in Parallel Lines and TransversalsAssessment Practice

4 marks~6 minCriterion A

The diagram shows two parallel lines cut by a transversal. One angle is labelled

65^\circ

.

a

State the size of the corresponding angle to the

65^\circ

angle. Justify your answer. [2]

b

A student claims: "The co-interior angle to $65^\circ$ must also be $65^\circ$ because it is in a matching position." Calculate the co-interior angle and explain why the student is incorrect. [2]

Solutions

24QuestionAngle Relationships in Parallel Lines and TransversalsAssessment Practice

4 marks~6 minCriterion D

A city planner designs a bridge crossing two parallel roads. The bridge acts as a transversal, forming a

72°

angle with the first road. Field measurements record the alternate interior angle on the second road as

70°

.

a

Calculate the angle the bridge should make with the second road if the roads are perfectly parallel. [1]

b

Explain why the

2°

discrepancy does not necessarily mean the roads are not parallel, referring to measurement error. [1]

c

Analyse one real-world factor that could cause the roads to deviate from perfect parallelism in practice. [2]

Solutions

Geometry – Properties of Shape — Free MYP3 Mathematics Practice Questions