# Angles, Polygons and Geometrical Proof - Stage 4

### Triangle Midpoints

##### Stage: 4 Challenge Level:

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

##### Stage: 4 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

##### Stage: 4 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

##### Stage: 4 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

##### Stage: 4 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

##### Stage: 4 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

##### Stage: 4 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

##### Stage: 4 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

##### Stage: 4 Challenge Level:

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

### Making Sixty

##### Stage: 4 Challenge Level:

Why does this fold create an angle of sixty degrees?

##### Stage: 4 Challenge Level:

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

### Kite in a Square

##### Stage: 4 Challenge Level:

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

### Overlapping Annuli

##### Stage: 4 Short Challenge Level:

Just from the diagram, can you work out the radius of the smaller circles?

### Circled Square

##### Stage: 4 Short Challenge Level:

Can you find the area of this square inside a circle?

### Overlapping Semicircles

##### Stage: 4 Short Challenge Level:

Two semicircles overlap, can you find this length?

### Octagonal Ratio

##### Stage: 4 Short Challenge Level:

Can you find the ratio of the area shaded in this regular octagon to the unshaded area?

### Block Challenge

##### Stage: 4 Short Challenge Level:

Can you work out the shaded area in this shape?

### Height and Sides

##### Stage: 4 Short Challenge Level:

Can you find the area of the triangle from its height and two sides?

### Circular Inscription

##### Stage: 4 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

##### Stage: 4 Short Challenge Level:

Can you find the radius of the circle inscribed inside a '3-4-5 triangle'?

### Slide Height

##### Stage: 4 Short Challenge Level:

How high is the top of the slide?

### Two Equilateral Triangles

##### Stage: 4 Short Challenge Level:

Prove that these two lengths are equal.

### Semicircle in a Triangle

##### Stage: 4 Short Challenge Level:

A semicircle is drawn inside a right-angled triangle. Find the distance marked on the diagram.

### Folded Square

##### Stage: 4 Short Challenge Level:

This square piece of paper has been folded and creased. Where does the crease meet the side AD?

### Tunnel Vision

##### Stage: 4 Short Challenge Level:

How wide is this tunnel?