problem

### Cyclic Quadrilaterals

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

problem

### Same length

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

problem

### circles in quadrilaterals

Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.

problem

### Isosceles Seven

Is it possible to find the angles in this rather special isosceles triangle?

problem

### Triangle midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

problem

### Two Ladders

Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second wall.
At what height do the ladders cross?

problem

### Quad in Quad

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

problem

### Sitting Pretty

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?

problem

### Trapezium Four

The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?

problem

### Nicely Similar

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?

problem

### Kite in a Square

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

problem

### The square under the hypotenuse

Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

problem

### Napkin

A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .

problem

### Angle Trisection

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

problem

### Squirty

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.

problem

### Partly Circles

What is the same and what is different about these circle
questions? What connections can you make?

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*ou may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.*