List

Angles, polygons and geometrical proof - Stage 4

Triangle midpoints
problem

Triangle midpoints

Age
14 to 16
Challenge level
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You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Two Ladders
problem

Two ladders

Age
14 to 16
Challenge level
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Two ladders are propped up against facing walls. At what height do the ladders cross?

Sitting Pretty
problem

Sitting pretty

Age
14 to 16
Challenge level
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A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Napkin
problem

Napkin

Age
14 to 16
Challenge level
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A napkin is folded so that a corner coincides with the midpoint of an opposite edge. Investigate the three triangles formed.

Angle Trisection
problem

Angle trisection

Age
14 to 16
Challenge level
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It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Squirty
problem

Squirty

Age
14 to 16
Challenge level
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Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

Trapezium Four
problem

Trapezium four

Age
14 to 16
Challenge level
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The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Nicely Similar
problem

Nicely similar

Age
14 to 16
Challenge level
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If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Partly Circles
problem

Partly circles

Age
14 to 16
Challenge level
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What is the same and what is different about these circle questions? What connections can you make?

Making sixty
problem

Making sixty

Age
14 to 16
Challenge level
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Why does this fold create an angle of sixty degrees?

circles in quadrilaterals
problem

Circles in quadrilaterals

Age
14 to 16
Challenge level
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Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Cyclic Quadrilaterals
problem

Cyclic quadrilaterals

Age
11 to 16
Challenge level
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Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Kite in a Square
problem

Kite in a square

Age
14 to 18
Challenge level
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Can you make sense of the three methods to work out what fraction of the total area is shaded?