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# The Art of Deduction

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Pentakite

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Quad in Quad

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Kite in a Square

Live
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The Converse of Pythagoras

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To Swim or to Run?

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### Making Sense of Statistics

### Integration as Area

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Creating convincing arguments or "proofs", to show that statements are always true, is a key mathematical skill. The problems in this feature offer you the chance to explore geometrical properties, make conjectures and create proofs to show that these are always
true.

Many of the problems in this feature include proof sorting activities which challenge you to rearrange statements in order to recreate clear, rigorous proofs.

Plus magazine has a selection of interesting articles exploring proofs in which pictures play an important role.

Age 14 to 18

Challenge Level

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

Age 14 to 18

Challenge Level

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Age 14 to 18

Challenge Level

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Age 14 to 18

Challenge Level

Can you prove that triangles are right-angled when $a^2+b^2=c^2$?

Age 16 to 18

Challenge Level

The famous film star Birkhoff Maclane wants to reach her refreshing drink. Should she run around the pool or swim across?

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of these resources.*

This collection of problems has been put together to help you to explore and understand important ideas in statistics.

These problems invite you to explore integration as area, and use area integrals to solve problems.