Triangle Midpoints
Age 14 to 16
Challenge Level
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
Two Ladders
Age 14 to 16
Challenge Level
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?
Sitting Pretty
Age 14 to 16
Challenge Level
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
Age 14 to 16
Challenge Level
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
Angle Trisection
Age 14 to 16
Challenge Level
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Squirty
Age 14 to 16
Challenge Level
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
Age 14 to 16
Challenge Level
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
Age 14 to 16
Challenge Level
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
Age 14 to 16
Challenge Level
What is the same and what is different about these circle questions? What connections can you make?
Circles in Quadrilaterals
Age 14 to 16
Challenge Level
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Cyclic Quadrilaterals
Age 11 to 16
Challenge Level
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Kite in a Square
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?
Quad in Quad
Age 14 to 18
Challenge Level
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
Angles, Polygons and Geometrical Proof Short Problems
Age 11 to 16
A collection of short problems on Angles, Polygons and Geometrical Proof.
Same Length
Age 11 to 16
Challenge Level
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
Isosceles Seven
Age 14 to 16
Challenge Level
Is it possible to find the angles in this rather special isosceles triangle?
The Square Under the Hypotenuse
Age 14 to 16
Challenge Level
Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?
Overlapping Annuli
Age 14 to 16 Short
Challenge Level
Just from the diagram, can you work out the radius of the smaller circles?
Circled Square
Age 14 to 16 Short
Challenge Level
Can you find the area of this square inside a circle?
Overlapping Semicircles
Age 14 to 16 Short
Challenge Level
Two semicircles overlap, can you find this length?
Octagonal Ratio
Age 14 to 16 Short
Challenge Level
Can you find the ratio of the area shaded in this regular octagon to the unshaded area?
Height and Sides
Age 14 to 16 Short
Challenge Level
Can you find the area of the triangle from its height and two sides?
Circular Inscription
Age 14 to 16 Short
Challenge Level
In the diagram, the radius of the circle is equal to the length AB. Can you find the size of angle ACB?
3-4-5 Circle
Age 14 to 16 Short
Challenge Level
Can you find the radius of the circle inscribed inside a '3-4-5 triangle'?
Triangular Intersection
Age 14 to 16 Short
Challenge Level
What is the largest number of intersection points that a triangle and a quadrilateral can have?
Semicircle in a Triangle
Age 14 to 16 Short
Challenge Level
A semicircle is drawn inside a right-angled triangle. Find the distance marked on the diagram.
Folded Square
Age 14 to 16 Short
Challenge Level
This square piece of paper has been folded and creased. Where does the crease meet the side AD?
Doubly Isosceles
Age 14 to 16 Short
Challenge Level
Find the missing distance in this diagram with two isosceles triangles
NRICH is part of the family of activities in the Millennium Mathematics Project.