(a) The diagram shows the position of town A and town B, on a map.
(i) Measure the length, in millimetres, of the line AB.
(ii) Measure the bearing of town B from town A.
(b) A triangular field has sides of length 550m, 300m and 400m.
(i) Construct the triangle, using a ruler and compasses only.
Use a scale of 1 cm to represent 50m.
The side of length 550m has been drawn for you.
(ii) By making a suitable measurement on your diagram, calculate the area of the field.
Give your answer in square metres.
▶️ Answer/Explanation
(a)(i) Ans: 44-46 mm
Using a ruler to measure AB on the diagram should give a length between 44-46 mm.
(a)(ii) Ans: 231°-235°
Using a protractor to measure the angle from north at A to the line AB should give a bearing between 231°-235°.
(b)(i) Ans: Fully correct drawing with arcs
Construction steps:
- Draw the base line (550m = 11 cm)
- From one end, draw an arc with radius 6 cm (300m scale)
- From other end, draw an arc with radius 8 cm (400m scale)
- The intersection point completes the triangle
(b)(ii) Ans: 52,250 to 60,500 m²
Method:
- Measure the height (h) from base to opposite vertex in cm
- Convert to meters: h × 50 (scale factor)
- Area = ½ × 550 × h = 275 × h
- Typical measurements yield areas between 52,250-60,500 m²
The diagram shows a plan, $ABCDE$, of the floor of a room in Jo’s house.
$F$ is a point inside the room.
(a)
(i) Show that $EF=1.9$m
(ii) Work out $AF$
(b) Calculate the area of the floor.
(c) A cupboard in the room is in the shape of a cuboid.
The area of the base of the cupboard is $1.2$ m² and the height of the cupboard is $2.3$ m.
Calculate the volume of the cupboard.
Give the units of your answer.
(d) Jo buys $275$ floor tiles which cost \$1.64 each.
Calculate the total cost of the floor tiles.
(e) Jo builds a patio in the shape of a semicircle with radius $2.3$ m.
Calculate the area of the patio.
▶️ Answer/Explanation
(a)(i) Proof:
$EF = DC – AB = 5.5\,m – 3.6\,m = 1.9\,m$
(a)(ii) Ans: 3 m
$AF = BC – ED = 4.7\,m – 1.7\,m = 3\,m$
(b) Ans: 23 m²
Total area calculation:
- Rectangle ABMC: $3.6 \times 4.7 = 16.92\,m²$
- Rectangle DEFM: $1.9 \times 1.7 = 3.23\,m²$
- Triangle AFE: $\frac{1}{2} \times 1.9 \times 3 = 2.85\,m²$
Total = $16.92 + 3.23 + 2.85 = 23\,m²$
(c) Ans: 2.76 m³
Volume = Base area × Height = $1.2 \times 2.3 = 2.76\,m³$
(d) Ans: \$451
Total cost = $275 \times 1.64 = \$451$
(e) Ans: 8.31 m²
Semicircle area = $\frac{1}{2} \times \pi \times (2.3)^2 = \frac{1}{2} \times \pi \times 5.29 ≈ 8.31\,m²$
Key Notes:
- For part (b), the floor area is calculated by dividing the shape into simple rectangles and a triangle
- All linear measurements are in meters (m), areas in square meters (m²), and volumes in cubic meters (m³)
- The semicircle area calculation uses exact value of π for precision
- Monetary values are rounded to the nearest dollar