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 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?

Tied Up

Age 14 to 16 Short
Challenge Level

How much of the field can the animals graze?

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?

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?

Square and Triangle

Age 14 to 16 Short
Challenge Level

Can you find the area of the yellow square?

Doubly Isosceles

Age 14 to 16 Short
Challenge Level

Find the missing distance in this diagram with two isosceles triangles

Circles on a Triangle

Age 14 to 16 Short
Challenge Level

Find the missing length on this diagram.

Cat on a Wall

Age 14 to 16 Short
Challenge Level

How high is the wall that this cat is lying on?