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A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

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Areas and Ratios

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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Six Discs

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Six circular discs, all of radius 1 unit, are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box.

  • Without using any calculations, compare the areas of the bases of boxes A and B and say which is the larger.
  • Use estimation to list the packings in order of increasing base area. Which are the hardest to put in order?
  • Now calculate the area of the base of each box and compare the correct order with your estimate.