Cambridge Essentials Mathematics Core 9 GM1.2 Answers

Cambridge Essentials Mathematics Core 9 GM1.2 Answers

1aPurple: A sector is the area trapped between two radii.

Orange: An arc is part of the edge of a circle.

Blue: The diameter is the distance from one side of the circle to the other, passing through the centre.

Green: The radius is the distance from the centre of the circle to the edge.

Grey: A tangent is a line that just touches the edge of the circle only once.

Red: A chord is a line that joins two points on the edge of a circle. If the chord passed through the origin, it is a diameter.

bThe circumference of the circlecThe radius

dIt only touches the circle at one point. It makes 90° with the radius.

2acircumscribed circlebinscribed circle

ccircumscribed circle dinscribed

3aEquilateral trianglesbAll the angles are 60°

c6 × 60° = 360°; so 6 triangles will fit exactly in the circle

dA regular hexagon

4First draw a circle with radius 4cm. Open a pair of compasses to a radius of 4cm andmark points round the edge of the circle that are all 4cm apart. Join these up to form the hexagon.

5ai360° ÷ 5 = 72° Use a protractor to draw 5 radii that have an angle of 72° between

them. Join the points on the edge of the circle to form five equal sides.

ii360° ÷ 7 = 51.4° Use a protractor to divide the circle into 7 equal parts by drawing a

radius every 51.4° . Join the points with straight lines to form a regular heptagon.

bCheck students’ pentagons andmeasuring. If accurate, all lengths will be the same.

cCheck students’ heptagons andmeasuring. If accurate, all lengths will be the same.

dPentagon: each side should be 5.88cm

Heptagon: each side should be 4.34cm

6aMultiply the length of AB by 18b18 × 2.1cm = 37.8cm

cDivide the circle into a greater number of equal parts, so making the length of the straight lines at the circumference shorter.

dC = π × 12 = 37.7cm This is slightly shorter than the estimated circumference

7aC = π × 8.5 = 26.7cm

bProviding that the diameter is accurate, then themost accurate answer is that given by using the formula and calculating.

8

91p coin 63.8mm

5p coin 56.5mm

10p coin 78.5mm

£1 coin 70.7mm

£2 coin 89.2mm

€1 coin 7.3cm (note units)

10C = πd is the same as C = 2πr because d is the same as 2r

11aDistance = circumference of tin = 23.2cm

bCircumference of golden syrup tin = 25.4cm, Gina’s tinmoves 2.2cm further.

12aC= π × 40 = 125.7cm = 1.257m, distance travelled = 1.257 × 1989 = 2500m = 2.5km

b1000 ÷ 1.257 = 795.5 revolutions or 1989 ÷ 2.5 = 795.6 revolutions

13a0.754mb0.754m × 106 = 79.9mc100.6 ÷ 0.754 = 133

14a4 + 8 × 0.35 + 4 × 0.05 = 7cm2b1.5cm

cA = π × 1.52 = 7.07cm2

dThe formula, A = πr2, it is quick and you do not need an accurate diagram.

eCheck that students’ reasons are sensible.

15r = radius, d = diameter, πd and 2πr are the formulae that work out the circumference of a circle, and πr2 works out the area

1643.0cm2 (to 1 d.p.)

17a201.06cm2b95.03cm2c145.27cm2d10.18cm2

18acheese cake = 254.47cm2bcookie = 63.62cm2

cpizza = 380.13cm2dbirthday cake = 346.36cm2

1966.0cm (to 1 d.p.)

20a490.9cm2 (to 1 d.p.)b78.5 + 1 = 79.5cm (to 1 d.p.)

22aFind the area of a circle with radius 2.5cm and halve the answer.

A = πr2 = π × 2.52 = 19.635cm2

Area of semicircle 19.635 ÷ 2 = 9.82cm2

bFind the perimeter of a circle with radius 2.5cm. Halve the answer and add on the diameter (5cm).

C = πd = π × 5 = 15.708, half circumference = 7.85cm

Perimeter of semicircle = 5 + 7.85 = 12.85cm

22a263m2b57.5m

23a 0.45m (to 2 d.p.)b0.64m2 (to 2 d.p.)

c(1.5 × 2.25) – 0.636 = 2.74m2 (to 2 d.p.)

24a1.77m2 (to 2 d.p.)bHalf of total area = 0.88m2 (to 2 d.p.)