Mathematics • Year 7 • Unit 3 • Lesson 19
Constructions — Real World
Apply construction ideas to real situations: finding the centre of a flowerbed, splitting a pie wedge fairly, dropping a perpendicular from a point onto a road, and building exact angles without a protractor.
1. Word problems
Each problem asks you to choose the right construction and describe it. You don't need to draw on this worksheet — just explain what you'd do with compass and straight-edge. Show your reasoning.
1.1 — Splitting a pizza fairly. Talia has a pie-slice shaped piece of pizza. She wants to cut it into two equal-angle pieces for herself and a friend, but she doesn't have a protractor.
(a) Which construction should she use?
(b) Describe the four steps in your own words. 3 marks
1.2 — Finding the centre of a garden bed. A rectangular garden bed has been marked out with stakes at corners A and B (one end of the long side). The gardener wants to find the exact midpoint of side AB to plant a centrepiece.
(a) Which construction finds the midpoint of AB?
(b) Why does the construction also guarantee the centrepiece line is at right angles to AB? 3 marks
1.3 — Mowing the shortest path to the road. Tomas is mowing the lawn. From a single tree (point T) he wants to mow the SHORTEST straight path to the edge of the road (line ℓ). The shortest path is always perpendicular.
(a) Which construction would he use?
(b) Why does the perpendicular give the SHORTEST path (and not, say, a slanted line)? 2 marks
1.4 — Making a 45° corner. An apprentice carpenter needs a 45° angle but has no protractor. Describe how she could construct an exact 45° angle using ONLY straight-edge and compasses (two constructions combined).
3 marks
1.5 — Tent guy ropes. A symmetric tent has its peak at point P, and the two guy ropes pull symmetrically away from P. The ground stake should be placed on the line that runs straight down from P, perpendicular to the ground line connecting the two rope ends (A and B).
(a) Which construction finds this line?
(b) Why does the line pass exactly under the tent peak? 2 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is doing the perpendicular bisector construction. They draw the first arc from A, then they SHIFT the compass to make it slightly bigger before drawing the arcs from B. They get a "perpendicular bisector" that doesn't actually pass through the midpoint of AB. In your own words, explain (i) WHY changing the radius breaks the construction, (ii) what the rule about the compass radius is in this construction, and (iii) what they should do to fix it.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Splitting the pizza
(a) The angle bisector construction.
(b) Steps: (i) Put compass point on the vertex of the slice (the tip). (ii) With any radius, draw an arc that crosses BOTH edges of the slice — call the crossings P and Q. (iii) Without changing the radius, draw an arc from P inside the slice. (iv) Same radius from Q — the two arcs cross at point R. (v) Use the straight-edge to draw the line from the vertex through R. That line bisects the angle of the slice into two equal halves.
1.2 — Garden bed midpoint
(a) The perpendicular bisector of AB.
(b) The perpendicular bisector passes through the midpoint of AB AND crosses AB at right angles (90°). That's what "perpendicular bisector" means — it's the unique line that is both perpendicular to AB and passes through its midpoint. So the centrepiece line is automatically at right angles to AB.
1.3 — Shortest path to road
(a) Perpendicular from a point to a line.
(b) The shortest distance from a point to a line is always along the perpendicular. Any slanted line from T to the road is the hypotenuse of a right triangle whose perpendicular side is shorter — so the perpendicular is the shortest path.
1.4 — Constructing 45°
Step 1: Start with a straight line and choose a point P on it. Construct the perpendicular at P using arcs (perpendicular bisector technique applied to a segment around P). This gives a 90° angle.
Step 2: Now bisect the 90° angle using the angle bisector construction. Each new angle = 90 ÷ 2 = 45°.
Two constructions combined: perpendicular + angle bisector.
1.5 — Tent guy ropes
(a) The perpendicular bisector of segment AB.
(b) Because the tent is symmetric, the peak P is equidistant from A and B (the two rope ends). Every point that is equidistant from A and B lies on the perpendicular bisector of AB. So P is ON that line, and the line passes straight down from P perpendicular to AB.
2.1 — Explain your thinking (sample response)
Changing the compass radius between drawing the arcs from A and the arcs from B breaks the construction because the perpendicular bisector relies on EQUIDISTANCE. The first arc says: "I am a fixed distance from A." The second arc says: "I am a fixed distance from B." When the two arcs cross, the crossing point is the same distance from both A and B — and any such point lies on the perpendicular bisector of AB. But if the radius changes between A's arcs and B's arcs, the crossing point is one distance from A and a DIFFERENT distance from B. That breaks equidistance, so the crossing is no longer on the perpendicular bisector. The rule is: do NOT touch the compass between drawing the A-arcs and the B-arcs. To fix it, my classmate should reset the radius (still wider than half of AB), redraw all four arcs without touching the compass between A and B, then join the new crossings.
Marking: 1 for naming equidistance; 1 for explaining how changing radius breaks it; 1 for stating the constant-radius rule; 1 for the practical fix.