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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

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?


Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?
The solution uses the formula for the area of a circle and gives practice in geometrical reasoning and manipulating algebraic expressions.

Possible approach
Start with a class discussion about the radii of the 4 semi-circles and the circle on AB as diameter. When the students have understood that there are only two unknowns involved, and the other radii can all be expressed in terms of the two unknowns, then they are ready to write down the areas and answer the question which they could do individually.

Dynamic geometry software could be a useful tool to aid investigation but this is not essential.

Key questions
If the blue semicircles have radii a and b what can you say about the other radii?

Possible support
Try Round and Round

Possible extension
Shaded Circles is a similar problem.

Learners could make up their own problems of this type.