Mathematics • Year 7 • Unit 3 • Lesson 20

Geometry Synthesis — Real World

Apply the full Unit 3 toolkit to real situations: stop signs, billboards, ramps, kite designs and orienteering courses. Mix shape facts, angle facts, parallel-line facts and similarity.

Apply · Real-World Maths

1. Word problems

Each problem mixes ideas from across Unit 3. Show your working with named reasons.

1.1 — Stop sign. A traffic stop sign is a regular octagon (8 equal sides, 8 equal interior angles).

(a) Find the interior angle sum and one interior angle.
(b) The sign is being enlarged for a new highway to be similar to the original, with a scale factor of 1.5. If the original side length is 40 cm, find the new side length and the new interior angle. 3 marks

Stuck on (b)? Similar means angles are PRESERVED — only sides scale. So the new interior angle = same as the original.

1.2 — Parallel railway tracks. Two railway tracks (parallel lines) cross a road at 75°. A small triangular sign sits below the tracks with one edge along the road, its top angle 75° (the same as where the road meets the tracks), and another angle of 60°.

(a) Use alternate angles to identify any angle equal to 75° inside the triangle.
(b) Find the third angle of the triangle, using ∠ sum of △. 3 marks

Stuck? Alternate angles on parallel lines (Z-shape) are equal. Then 180 − 75 − 60 for the third angle.

1.3 — Wheelchair ramp + drop. A wheelchair ramp from the pavement to the door is similar to a smaller mock-up: the model has length 25 cm and height 5 cm; the real ramp has length 200 cm.

(a) Find the scale factor from model to real.
(b) Find the real ramp's height.
(c) State (without calculating) why the ramp's angle with the ground is the same for both the model and the real ramp. 3 marks

Stuck on (c)? Similar figures have equal corresponding ANGLES — only sides scale.

1.4 — Kite design. A toy kite ABCD is shaped like a geometric kite: AB = AD (two short equal sides) and CB = CD (two long equal sides). The angle at the top vertex A is 80° and the angle at one side vertex (B) is 110°.

(a) State the size of angle D, with reason.
(b) Find angle C (the bottom vertex). 3 marks

Stuck on (a)? A geometric kite has a line of symmetry from A to C — so ∠B = ∠D = 110°. For (b): quadrilateral angle sum = 360°.

1.5 — Orienteering course. An orienteering map shows three checkpoints forming a triangle PQR. The map scale is 1 : 2500. On the map, PQ = 8 cm, QR = 12 cm, PR = 10 cm.

(a) Find the REAL distances from P to Q, Q to R and P to R, in metres.
(b) What scale factor takes the map triangle to the real triangle? 3 marks

Stuck? Multiply each map distance (in cm) by 2500 to get real distance in cm, then ÷ 100 for metres. The scale factor IS 2500 (real ÷ map).

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate looks at a square and says: "It can't be a rectangle and a rhombus AND a parallelogram all at the same time — it has to be just ONE of them." In your own words, explain (i) what each of these four quadrilateral names actually requires (square, rectangle, rhombus, parallelogram), (ii) why a square satisfies ALL FOUR definitions at once, and (iii) why we usually call it a "square" (the most specific name) rather than a "rectangle" or "parallelogram".

Stuck? Revisit lesson § Card 5 "Shape Properties Snapshot" — a square is the most special quadrilateral and inherits all the properties of rectangles, rhombi and parallelograms.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Stop sign (regular octagon)

(a) Sum = (8 − 2) × 180 = 1080°. Each angle = 1080 ÷ 8 = 135°.
(b) New side = 40 × 1.5 = 60 cm. Interior angle = 135° (unchanged — similarity preserves angles).

1.2 — Parallel railway tracks

(a) The 75° at the road–track intersection equals an angle inside the triangle (alt ∠s, tracks parallel). One angle of the triangle = 75°.
(b) Third angle = 180 − 75 − 60 = 45° (∠ sum of △).

1.3 — Wheelchair ramp

(a) SF (model → real) = 200 ÷ 25 = 8.
(b) Real height = 5 × 8 = 40 cm.
(c) Similar figures have equal corresponding angles — only the sides scale. The ramp angle is determined by the proportion of height to length, which is the SAME (5/25 = 40/200) for both the model and the real ramp.

1.4 — Kite design

(a) ∠D = 110° (geometric kite is symmetric about the diagonal AC, so ∠B = ∠D).
(b) Quadrilateral angle sum: ∠A + ∠B + ∠C + ∠D = 360°. So ∠C = 360 − 80 − 110 − 110 = 60°.

1.5 — Orienteering course

(a) PQ real = 8 × 2500 = 20 000 cm = 200 m.
QR real = 12 × 2500 = 30 000 cm = 300 m.
PR real = 10 × 2500 = 25 000 cm = 250 m.
(b) Scale factor = real ÷ map = 2500.

2.1 — Explain your thinking (sample response)

A square is a special case of all three other shapes — it satisfies every definition at once. Here's what each name requires: parallelogram — two pairs of parallel sides; rectangle — four right angles (which means it's also a parallelogram); rhombus — four equal sides (which means it's also a parallelogram); square — four equal sides AND four right angles. A square has both equal sides (so it's a rhombus) AND right angles (so it's a rectangle), and since both rectangles and rhombi have parallel opposite sides, a square is also a parallelogram. We call it a "square" because that's the MOST SPECIFIC name — it carries the most information. Calling a square a "parallelogram" is technically correct but vague, like calling a labrador a "mammal". When all the more specific properties are true, use the most specific name.

Marking: 1 mark for defining each of square / rectangle / rhombus / parallelogram correctly; 1 mark for explaining why square satisfies all four; 1 mark for "most specific name" justification; 1 mark for clear full-sentence communication.