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Squaring a rectangle · 4 marks

Try to work out why this method works. You will find that it is a geometrical translation of the formula (a + b2)2 − (a − b2)2 = ab.

ADBCEFGHKPQSab
Answer: AF = a + b2, so HK = HG = a + b2, and KP = BH = AF − AB = a − b2. In right triangle HPK: HP2 = HK2 − KP2 = (a + b2)2 − (a − b2)2 = ab. So square HPQS has the area of the rectangle.

Step-by-step solution

Idea: The side of the big square AFGH is the average a + b2, and the arc turns it into the hypotenuse of a right triangle whose other side is a − b2.

  1. AE = AB = b, and F is the midpoint of ED, so AF = AE + AD2 = a + b2. AFGH is a square, so HG = HA = a + b2, and HK = HG (radii of the arc).1 mark
  2. BH = AH − AB = a + b2 − b = a − b2. BKPH is a rectangle (KP ∥ AH), so KP = BH = a − b2.1 mark
  3. Triangle HPK is right-angled at P. Baudhāyana–Pythagoras: HP2 = HK2 − KP2 = (a + b2)2 − (a − b2)2.1 mark
  4. = a2 + 2ab + b24 − a2 − 2ab + b24 = 4ab4 = ab. So the square HPQS (area HP2) equals the rectangle ABCD (area ab).1 mark
HP² = HK² − KP² = ((a + b)/2)² − ((a − b)/2)² = ab, so square HPQS has the same area as rectangle ABCD.

Check: With a = 9, b = 4 (as in the picture): AF = 6.5, BH = 2.5, HP = √(6.52 − 2.52) = √36 = 6, and 62 = 36 = 9 × 4 ✓.

Answer to write in the exam

AF = AE + AD2 = a + b2 ⇒ HK = HG = a + b2 (radii)

KP = BH = AH − AB = a + b2 − b = a − b2

In right △HPK: HP2 = HK2 − KP2

= (a + b2)2 − (a − b2)2 = 4ab4 = ab

∴ ar(HPQS) = ar(ABCD)

Common mistakes that cost marks

  • Taking AF = a2. F is the midpoint of ED, not of AD.
  • Using HP2 = HK2 + KP2. HK is the hypotenuse, so subtract.
  • Thinking KP equals AB; it equals BH.

How this can come in the exam

MCQ (1 mark)

Using (a + b2)2 − (a − b2)2 = ab, a rectangle 32 cm × 8 cm has the same area as a square of side

  1. 8 cm
  2. 12 cm
  3. 20 cm
  4. 16 cm
Show answer

(D) 16 cm
202 − 122 = 400 − 144 = 256 = 32 × 8, so the side is √256 = 16 cm.

Try one yourself

For a rectangle 25 cm × 9 cm, find AF, BH and the side of the square in Baudhāyana’s construction.

Show answer

AF = 17, BH = 8, side = √(172 − 82) = √225 = 15 cm (152 = 225 = 25 × 9).

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