Mathematics • Year 7 • Unit 3 • Lesson 8

Parallelograms and Rectangles — Real World

Use parallelogram and rectangle properties to find angles and lengths in tabletops, doors, basketball courts, gate frames and bookshelves. Always state the property you used in brackets after each line.

Apply · Real-World Maths

1. Word problems

Read carefully. For each scenario, identify the shape, choose the property, and write the reason in brackets.

1.1 — Basketball court. A basketball court is a rectangle 28 m long and 15 m wide. Diagonal lines are painted from corner to corner, meeting at the centre circle.

(a) Are the two diagonal lines the same length? Why?
(b) Do they cut each other in half at the centre? Why? 2 marks

Stuck? Diagonals of a rectangle are equal AND bisect each other.

1.2 — Garden gate frame. A garden gate is built as a parallelogram (it slants slightly when open). The top-left corner angle is 75°.

(a) Find the top-right corner angle.
(b) Find the angle at the bottom-right corner.
(c) State the property you used for each. 3 marks

Stuck? Top-right is co-interior with top-left (180°). Bottom-right is opposite top-left (equal).

1.3 — Bookshelf side panel. The side panel of a bookshelf is a parallelogram PQRS. PQ = 1.8 m and QR = 0.4 m.

(a) Find SR and PS using the opposite-sides-equal property.
(b) State the property in brackets. 2 marks

Stuck? PQ is opposite SR; QR is opposite PS.

1.4 — Door diagonal. A rectangular door is 2.1 m tall and 0.9 m wide. Its two diagonal braces meet at the middle of the door.

(a) Without computing the actual diagonal length, explain why the two diagonal braces are the same length.
(b) Each brace is cut in half by the centre point. If the full diagonal turns out to be 2.285 m (rounded), what is the distance from the centre to each corner? 2 marks

Stuck on (a)? Diagonals of a RECTANGLE are equal (parallelograms alone aren't enough).

1.5 — Tabletop survey. A carpenter measures a parallelogram-shaped tabletop ABCD and finds AB = 120 cm and ∠A = 100°. The opposite side DC is reported as 118 cm.

(a) What should DC actually equal if the shape is a true parallelogram? Why?
(b) The carpenter rechecks and confirms ∠C = 100°. Find ∠B and ∠D. 3 marks

Stuck? In a true parallelogram, opposite sides are equal, so DC should equal AB exactly.

2. Explain your thinking

Use full sentences and reference specific properties. 4 marks

2.1 A classmate looks at a parallelogram and says "the diagonals must be equal in length because they bisect each other." Explain (i) why "bisect" is NOT the same as "equal in length", (ii) the parallelogram WHEN diagonals are equal (give the name), and (iii) which everyday object in your classroom you'd use to demonstrate the difference.

Stuck? "Bisect" means cut in half. Two segments can be cut in half at the same point without being equal in length. Equal diagonals only appear in a rectangle.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Basketball court diagonals

(a) Yes, equal in length. A basketball court is a rectangle, and diagonals of a rectangle are equal in length (diagonals of rectangle equal).
(b) Yes, they cut each other in half. Every rectangle is also a parallelogram, and the diagonals of a parallelogram bisect each other (diagonals of parallelogram bisect).

1.2 — Garden gate, top-left ∠ = 75°

(a) Top-right = 180 − 75 = 105° (co-int. angles, top side ∥ bottom side).
(b) Bottom-right = opposite the top-left, so = 75° (opp. angles of parallelogram).
(c) Properties used: co-interior angles sum 180°, and opposite angles of a parallelogram are equal.

1.3 — Bookshelf, PQ = 1.8, QR = 0.4

(a) SR = PQ = 1.8 m, and PS = QR = 0.4 m.
(b) Reason: opp. sides of parallelogram (equal).

1.4 — Door diagonals, 2.1 × 0.9 rectangle

(a) The door is a rectangle, and diagonals of a rectangle are equal in length (diagonals of rectangle equal), so the two braces are the same length.
(b) Half-diagonal = 2.285 ÷ 2 ≈ 1.143 m (or about 114 cm) from the centre to each corner (diagonals bisect each other).

1.5 — Tabletop ABCD

(a) In a true parallelogram, opposite sides are equal, so DC should equal AB exactly = 120 cm. The measurement of 118 cm suggests the tabletop has been built slightly off (or measured imprecisely) — it isn't quite a true parallelogram (opp. sides of parallelogram).
(b) ∠C = 100° = ∠A confirms opposite angles equal. ∠B = 180 − 100 = 80° (co-int. angles, AB ∥ DC). ∠D = ∠B = 80° (opp. angles). Check: 100 + 80 + 100 + 80 = 360° ✓.

2.1 — Explain your thinking (sample response)

(i) "Bisect" means "cut into two equal halves" — that's about each diagonal cutting the OTHER one in half at the centre. It does NOT mean the two diagonals are the same length as each other. A long diagonal can still be bisected at its middle, and a separate short diagonal can also be bisected at the same middle point — they just happen to be different total lengths.
(ii) Diagonals are equal in length only when the parallelogram is a rectangle.
(iii) I'd use a non-rectangular parallelogram object like a slanted picture frame or a tilted whiteboard tile. Drawing both diagonals shows the two cross at the centre, but one diagonal is clearly longer than the other.

Marking: 1 for distinguishing "bisect" vs "equal in length"; 1 for naming rectangle; 1 for a sensible example; 1 for clear full-sentence explanation.