What fraction of the square is shaded?
Step-by-step solution
Idea: The lines come in two parallel pairs, so the converse of the Midpoint Theorem cuts each line into equal pieces of length s (the side of the inner square). A right triangle with legs 2s and s then has the big square’s side as hypotenuse.
- Name the square ABCD with midpoints K, L, M, N of AB, BC, CD, DA; the lines are AM, KC, DL, NB, and they enclose PQRS. AKCM and NBLD are parallelograms (AK = MC, AK ‖ MC; similarly), so AM ‖ KC and DL ‖ NB. Also ∆ADM ≅ ∆DCL (SAS), so ∠DAM = ∠CDL, giving ∠APD = 90°: AM ⟂ DL. So PQRS has four right angles.1 mark
- In ∆DCQ, M is the midpoint of DC and MP ‖ CQ, so P is the midpoint of DQ: DP = PQ. In ∆CBR, L is the midpoint of CB and LQ ‖ BR, so Q is the midpoint of CR: CQ = QR.1 mark
- ∆DQC ≅ ∆CRB (AAS: right angles at Q and R, DC = CB, ∠QDC = ∠RCB from ∆DCL ≅ ∆CBK), so DQ = CR, i.e. 2PQ = 2QR. So PQ = QR = QC = s and DQ = 2s; PQRS is a square of side s.1 mark
- Right ∆DQC: DC2 = DQ2 + QC2 = 4s2 + s2 = 5s2. Shaded fraction = s25s2 = 15.1 mark
Check: Coordinates with side 2: D(0, 0), C(2, 0), B(2, 2), A(0, 2). The inner square has corners (0.8, 0.4), (1.6, 0.8), (1.2, 1.6), (0.4, 1.2); its side is √(0.64 + 0.16) = √0.8, area 0.8 = ⅕ × 4 ✓.
Answer to write in the exam
AM ‖ KC, DL ‖ NB (AKCM, NBLD parallelograms); ∆ADM ≅ ∆DCL (SAS) ⇒ AM ⟂ DL ⇒ PQRS has right angles
∆DCQ: M midpoint of DC, MP ‖ CQ ⇒ DP = PQ; ∆CBR: L midpoint of CB, LQ ‖ BR ⇒ CQ = QR
∆DQC ≅ ∆CRB (AAS) ⇒ DQ = CR ⇒ PQ = QR = QC = s, DQ = 2s
DC² = DQ² + QC² = 4s² + s² = 5s²
∴ Shaded fraction = s²/5s² = 1/5
Common mistakes that cost marks
- Guessing ¼ because the lines go to midpoints. The inner square is smaller: ⅕.
- Assuming DP = PQ = QL (three equal parts); actually QL is only half of PQ.
- Forgetting to show the middle figure is a square before squaring its side.
How this can come in the exam
In the figure, the big square has side 10 cm. The area of the shaded square is
- 10 cm²
- 20 cm²
- 25 cm²
- 40 cm²
Show answer
(B) 20 cm²
Shaded area = ⅕ × 100 = 20 cm².
Try one yourself
The shaded square in this figure has area 9 cm². Find the side of the big square.
Show answer
Big square area = 5 × 9 = 45 cm², so its side is √45 = 3√5 cm.
More questions like this
- Can we use any given quadrilateral to tile the plane? If not, which quadrilaterals can be used and which cannot?
- Informally, a quadrilateral is a figure with four straight sides, as in the first figure ABCD in the figure below. But consider the other six figures in the figure: the five plane figures NOPE, SILY, DART, CUTS, OPENS and the non-planar BENT. Should we call all these figures quadrilaterals? If you answer ‘no’ for any of them, how will you define a quadrilateral so that such a figure is excluded? As you can see, some care is needed to precisely define what we think of as a quadrilateral.
- To prepare, let us first consider how we can define a triangle. Let A, B and C be three points. Can we say that ∆ABC consists of points on the three line segments AB, BC, CA?
- Can we similarly define a quadrilateral ABCD?
- Take any 4 distinct non-collinear points A, B, C and D. Will segments AB, BC, CD and DA always form a quadrilateral as we visualise it?
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