Angles, Polygons and Geometrical Proof - Stage 4

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?

Making Sixty

Age 14 to 16 Challenge Level:

Why does this fold create an angle of sixty degrees?

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 16 Challenge Level:

Can you make sense of the three methods to work out the area of the kite in the square?

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?

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?

Block Challenge

Age 14 to 16 Short Challenge Level:

Can you work out the shaded area in this shape?

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'?

Slide Height

Age 14 to 16 Short Challenge Level:

How high is the top of the slide?

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?

Two Equilateral Triangles

Age 14 to 16 Short Challenge Level:

Prove that these two lengths are equal.

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?

Tunnel Vision

Age 14 to 16 Short Challenge Level:

How wide is this tunnel?

Doubly Isosceles

Age 14 to 16 Short Challenge Level:

Find the missing distance in this diagram with two isosceles triangles