Y9 – Autumn – Block 5 – Step 1 – Draw and measure angles Answers

Question Answer

1

a) The angle is larger than a right angle, so must be more than 90°.b) Nijah has used the wrong scale on the protractor. She needs to make sure she starts

from 0 degrees.c) 150°

2

Accept 2° either sidea) 48°b) 126°c) 26°d) 148°

3

Correct angle drawn.

or

4

Correct diagram drawn.

5

Correct angles drawn.a)

b)

6 Work out 225 − 180 = 45, and draw an angle of 45°. The reflex angle is 225°.

P

R Q

X Y

Z

Y9 – Autumn – Block 5 – Step 1 – Draw and measure angles Answers (continued)

Question Answer

7

Accept 2° either sidea) 200°b) 296°c) 232°d) 314°e) 270°f) 218°

8

Correct angles drawn. E.g.a)

b)

c)

9

3.6 cm (Accept 2 mm either side)

Y9 – Autumn – Block 5 – Step 2 – Construct and interpret scale drawings Answers

Question Answer

1

a) 4 mb) 6 mc) 0.5 md) 3.3 m

2

a)

b) 32 squares by 20 squares

3

a) 1 mb) 1.5 m by 1 mc) 0.7 m by 0.4 md)

4

All distances in miles. Accept 6 miles either side

Please note: Answer might vary slightly depending on print settings.

5

6

a) Always true. For a scale of 1 : 20,000, a 200 m line is represented by 1 cm. For a scale of 1 : 40,000, a 200 m line is represented by 0.5 cm

b) Never true. 1 cm represents 1 km is a scale of 1 : 1 × 100 × 1,000 = 1 : 100,000c) Always true. Any distance on the actual object is multiplied by the scale factor for the

length on the drawing, so the drawing lengths are all greater than the actual lengths.

1 cm

0.8 cm

10510884

189117

105198

84198

108189 117

Y9 – Autumn – Block 5 – Step 3 – Locus of distance from a point Answers

Question Answer

1

a) Circle of radius 3 cm drawn.

b) Circle of radius 2.5 cm drawn.

2

a) Circle of radius 4 cm drawn.

b) 25.1 m

Y9 – Autumn – Block 5 – Step 3 – Locus of distance from a point Answers (continued)

Question Answer

3

a) All parts of circles drawn should have radius 3.5 cm.

b) Goat A can graze the most grass, as the area is a semicircle. The other areas are less than a semicircle.

Please note: Answer might look different depending on print settings.

Y9 – Autumn – Block 5 – Step 3 – Locus of distance from a point Answers (continued)

Question Answer

4

a) The unshaded area shows where it is not possible to access the internet.

b) multiple possible answersStudents need to choose two access points to maximise the area covered by two circles with radius 5 cm. It is not possible to cover the whole house. They could discuss whether it is more important to cover the utility room or the dining room. Also, they may consider where access points can be positioned, for example on walls.

Please note: Answer might look different depending on print settings.

5

Two circles drawn.One has radius 2 cm and the other has radius 3.5 cm.The space between them should be shaded.

Y9 – Autumn – Block 5 – Step 3 – Locus of distance from a point Answers (continued)

Question Answer

6

This is slightly reduced from actual size, but the relative positions of the circle arcs should look the same.

c) It is not possible to plant one tree that matches both criteria as the two areas do not overlap.

Please note: Answer might look different depending on print settings.

a)

b)

Y9 – Autumn – Block 5 – Step 4 – Locus of distance from a straight line or shape Answers

Question Answer

1

WhitneyThe locus from the end of each line is a semicircle. The corners of Annie’s rectangle are further from the line than 2 cm.

2

All points in the locus should be 3 cm away from the line segment.

3

Correct locus drawn.

4

a)

Y9 – Autumn – Block 5 – Step 4 – Locus of distance from a straight line or shape Answers (continued)

Question Answer

4

b)

c)

d)

All the loci have arc sections or a circle with radius 2 cm. All except the circle shape also have straight line sections.The positions of the arcs and straight lines depend on the original shape.

Y9 – Autumn – Block 5 – Step 4 – Locus of distance from a straight line or shape Answers (continued)

Question Answer

5

Correct region shaded.

Please note: Answer might look slightly different depending on print settings.

6

a)

b) Answer is between 35.2 cm2 and 41.0 cm2 when printed on A4.Please note: Answer will vary slightly depending on print settings.

Y9 – Autumn – Block 5 – Step 4 – Locus of distance from a straight line or shape Answers (continued)

Question Answer

7

Correct region shaded.

Please note: Answer might look slightly different depending on print settings.

8

a) Correct locus drawn.

b) Answer is between 38.2 cm2 and 40.6 cm2 when printed on A4.Please note: Answer will vary slightly depending on print settings.

Y9 – Autumn – Block 5 – Step 5 – Locus equidistant from two points Answers

Question Answer

1

a) Students should measure the distance between the cross and each star and see that the distances are the same.

b) multiple possible answers, e.g.:

2

a) 5 cmb)

c) The points where circles of the same radius intersect are equidistant from X and Y.

3

a) Correct locus drawn.

b) Correct locus drawn.

X Y

Y9 – Autumn – Block 5 – Step 5 – Locus equidistant from two points Answers (continued)

Answer

4b)

Please note: Answer might look slightly different depending on print settings.

5

Y9 – Autumn – Block 5 – Step 5 – Locus equidistant from two points Answers (continued)

Question Answer

6

7

a) The towns might not be in opposite directions from Ron. For Ron to be exactly halfway between the two towns, he needs to be on the straight line joining the two towns.

b) A circle of radius 4 cm drawn using Ron as centre.

Y9 – Autumn – Block 5 – Step 6 – Construct a perpendicular bisector Answers

Question Answer

1

2A perpendicular bisector is a line that is at right angles to the original line and divides it into two equal parts.

3

Correct perpendicular bisector drawn.

✔ ✔

Question Answer

4

Correct perpendicular bisector drawn for each part.

Y9 – Autumn – Block 5 – Step 6 – Construct a perpendicular bisector Answers (continued)

Question Answer

5

Filip has not use a large enough radius for the arcs centred on the ends of the line segment. The radius needs to be large enough for the arcs to intersect above and below the line segment.

6

a) Line of length 6.4 cm drawn.b) Correct perpendicular bisector drawn.c) Point Z correctly labelled.

d) 3.2 cmThere is no need to measure XZ because it is half the length of XY.

7

Correct construction drawn.

8

Set compasses to a radius greater than half the distance between the points.Draw a circle (or arc of a circle) centred on each end of the line, using this radius.Draw the straight line that passes through both points of intersection of the circles (or arcs).

Y9 – Autumn – Block 5 – Step 6 – Construct a perpendicular bisector Answers (continued)

X YZ

Y9 – Autumn – Block 5 – Step 7 – Construct a perpendicular from a point Answers

Question Answer

1

a) Correct perpendicular drawn for each.

b) Both methods work because the points where Dani’s circle and Huan’s arcs cross the line segment are equidistant from point P.

2

Correct construction.

Y9 – Autumn – Block 5 – Step 7 – Construct a perpendicular from a point Answers (continued)

Question Answer

3

Correct construction for each part.a)

b)

c)

4

a)

b) 40 cm2

B

Y9 – Autumn – Block 5 – Step 7 – Construct a perpendicular from a point Answers (continued)

Question Answer

5

a) XY = 3.6 cmYZ = 5 cmZX = 5 cm

b)

c) 8.3 cm2 (accept 1 cm2 either side)

Please note: Answers may vary slightly depending on print settings.

6

Step 1: Draw a circle centred on the point. It is important that the circle intersects the line segment in two places.Step 2: Mark the points where the circle intersects with the line segments. These are two points equidistant from the original point.Step 3: Draw two circles with the same radius centred on the two points in step 2 on the line segment. These will give two points of intersection on a line perpendicular to the line segment.Step 4: Draw the line that passes through the intersection of the two circles in step 3 and also through the original point.

The same construction could be done just drawing arcs rather than complete circles or by drawing the original circle centred anyway vertically above or below the point as long as it intersects the line at two points.

Y9 – Autumn – Block 5 – Step 8 – Construct a perpendicular to a point Answers

Question Answer

1

Correct construction in each part.a)

b)

c)

2

Correct construction in each part.a)

b)

Both methods involve finding two points on the line segment that are equidistant from the point.When the point is on the line, the compasses must then be opened out to a greater radius to draw two arcs that intersect above and below the line segment. For the point that is not on the line, the same radius can be used to draw two arcs the other side of the line segment from the point.

Y9 – Autumn – Block 5 – Step 8 – Construct a perpendicular to a point Answers (continued)

Question Answer

3

The correct methods all involve finding two points equidistant from the point on the line segment and then drawing two circles or arcs with the same radius centred on those two points.

4

a), b), c), d) correctly constructed as shown.

3.6 cm (accept 0.5 cm either side)

5

XZ should be 2 cm and ZY should be 8 cm long.

✔ ✔

X YP

Q

X YZ

Y9 – Autumn – Block 5 – Step 8 – Construct a perpendicular to a point Answers (continued)

Question Answer

6

There are various ways of constructing a square. For the method shown:• Draw a long line.• Mark two points 6 cm apart in the middle of the line.• Construct the perpendiculars through the points.• Measure 6cm from the line up each perpendicular.• Join the points.

7

Use the same radius to draw an arc each side of the point centred on the point.Increase the radius of the compasses.From each point where the arcs intersected the line segment, draw an arc above and below the line.Draw the line that passes through the intersections of the arcs and the point on the line segment.

Y9 – Autumn – Block 5 – Step 9 – Locus of distance from two lines Answers

Question Answer

1

For each part, any three points drawn on the line shown.

a) c)

b) d)

2

a) Any three points marked on the line shown.

b) Angle ABD = 27°Angle CBD = 27°The two angles are equal. The locus of points equidistant from AB and BC have cut angle ABC exactly in half (or bisected the angle).

Y9 – Autumn – Block 5 – Step 9 – Locus of distance from two lines Answers (continued)

Question Answer

3

4

5

Please note: Answer might look slightly different depending on print settings.

Y9 – Autumn – Block 5 – Step 9 – Locus of distance from two lines Answers (continued)

Question Answer

6

a)

b) 2.4 km (accept 0.2 km either side)c) Rosie does not come within 1 km of the pond.

The locus of points 1 km from the pond does not reach the path that Rosie runs on.

Please note: Answers might vary slightly depending on print settings.

Y9 – Autumn – Block 5 – Step 10 – Construct an angle bisector Answers

Question Answer

1 ‘Bisect’ means to divide into two equal parts.

2

a)

b) In the second diagram, the angle is not divided into two equal parts.In the third diagram, the angle is divided into three equal parts, not two equal parts.

c) An angle bisector divides an angle into two equal angles.

3 Students need to justify their choice of method.

4

a)

b)

c)

d)

✔

Y9 – Autumn – Block 5 – Step 10 – Construct an angle bisector Answers (continued)

Question Answer

5

a)

b)

c)

d)

The angles each side of the angle bisector are equal.

6 No.The angles are not equal.

7

Set the compasses to a length less than the shorter of the two arms making the angle.Draw arcs with this radius centred on the angle so that they intersect with both arms of the angle.Draw two arcs centred on where the previous arcs intersected with the arms of the angle. Make sure that the arcs intersect.Draw a line through the angle and the intersection of the arcs.

Y9 – Autumn – Block 5 – Step 11 – Construct triangles from given information Answers

Question Answer

1

Correct triangle drawn in each part.

b)

c)

d)

2

a)

b) 7.5 cm (accept 0.5 cm either side)Q R

P

Y9 – Autumn – Block 5 – Step 11 – Construct triangles from given information Answers (continued)

Question Answer

3

It is more efficient to draw the 6 cm side first, as then the compasses only need to be set to 5 cm to draw the arcs.

4

a)

b) The sum of the two shorter sides is less than the longest side, so these two sides can never meet.

c)

✔

✔

Y9 – Autumn – Block 5 – Step 11 – Construct triangles from given information Answers (continued)

Question Answer

5

6

Y9 – Autumn – Block 5 – Step 12 – Identify congruent figures Answers

Question Answer

1

a) Jack is correct. We need information about sides and angles to be sure that shapes are congruent.

b) She could measure the sides and angles.

2a) Congruent means that two shapes are identical, with the same sides and angles.b) D and F

3

a) D and Gb) four possible answers, in different orientations, e.g.:

c) eight possible answers, in different orientations and reflections, e.g.:

4

a) sometimes trueTwo different shapes, for examples a square and a triangle, can have the same area but are not congruent. But if two shapes are congruent they must have the same area.

b) always trueIf two shapes are identical, they have the same dimensions and so must have the same area.

5

a)

b) Yesc) Yes. Each vertex is translated the same amount, so the image is the same shape as the

object.

6trueThe image has the same sides and angles as the object.

7Yes. The circles are identical, because if a circle has a radius of 3.6 cm then its diameter must be 7.2 cm.

8 The shapes have the same sides and angles. They also have the same area and perimeter.

9 1 : 1

B

Y9 – Autumn – Block 5 – Step 12 – Identify congruent figures Answers (continued)

Question Answer

10

width of each rectangle = !" of 8 cm = 4 cm

area of each rectangle 4 × 8 = 32 cm2

total area = 3 × 32 = 96 cm2

Y9 – Autumn – Block 5 – Step 13 – Explore congruent triangles Answers

Question Answer

1 The sides and angles of the triangles are the same, but the orientation is different.

2

a) multiple possible answers, e.g.:

b) Students need to check that the triangles have the same three sides to be congruent.

3

a)

b)

c)

The triangles in parts a) and c) should be congruent, but the triangles in part b) are unlikely to be congruent, as a triangle can have the same angles but be any size.

Y9 – Autumn – Block 5 – Step 13 – Explore congruent triangles Answers (continued)

Question Answer

4

a) Amir is correct.If the sides are the same, the angles must also be the same.

b) Kim is incorrect.The angles can be the same, but the sides may be different. One triangle will be an enlargement of the other.

5

SSS: three sides the sameASA: two angles the same and a corresponding side the sameSAS: two sides and the angle between them the sameRHS: a right angle and the hypotenuse and one other side the same

6

multiple possible answers, e.g.:

7 42 cm

5 cm

50°

60° 70°

5 cm

50°

60° 70°

Y9 – Autumn – Block 5 – Step 14 – Identify congruent triangles Answers

Question Answer

1

a) not enough informationb) congruent c) congruentd) congruente) congruentf) congruentStudents could discuss their reasons in terms of the conditions for congruency:a) could be any sizeb) RHSc) ASAd) SAS or ASAe) SASf) SAS or ASA

2

multiple possible answers, e.g.:

ASA

3

B is not congruent because the 9 cm side opposite a different angleC: SASD: SSSE: ASA

4

DIn triangle A, the top angle = 180 − 2 × 70 = 40°So triangle D has the same two sides of 17 cm with an angle of 40° between them.

5B and CThe angle of 52° is between the two sides of 7 cm and 8 cm.

5 cm50°

70°5 cm

50° 70°

✔

✔ ✔

Y9 – Autumn – Block 5 – Step 14 – Identify congruent triangles Answers (continued)

Question Answer

6

Maybe. There are two possible triangles:

7 18 cm2

8

Yes.Students can use different properties of parallelograms to prove congruency, e.g.:AD = BC and AB = DC (opposite sides are equal)AC is common to both trianglesSo triangles ABC and CDA are congruent (SSS)

angle DAC = angle BCA (alternate angles)angle ACD = angle CAB (alternate angles)AC is common to both trianglesSo triangles ABC and CDA are congruent (ASA)

angle DAC = angle BCA (alternate angles)angle ADC = angle CBA (opposite angles are equal)AC is common to both trianglesSo triangles ABC and CDA are congruent (ASA)

9

a) sometimes trueThe angles are the same, but the sides can be any length. The triangles are only congruent if they have the same side length.

b) sometimes trueThe angles are the same, but the triangle can be any size. The triangles are only congruent if the sides are also the same.

c) sometimes trueTwo congruent triangles will have the same perimeter, but two triangles can have different length sides that give the same perimeter, e.g. an equilateral triangle with side 4 cm and an isosceles triangle with sides 5 cm, 5 cm and 2 cm.

d) sometimes trueTwo congruent triangles will have the same area, but two triangles can be different and still have the same area, e.g. a triangle with base 4 cm and height 6 cm will have the same area as a triangle with base 8cm and height 3 cm.

A B

A B