# Angles, Triangles and Trigonometry - June 2007, All Stages

## Problems

### Turning

##### Stage: 1 Challenge Level:

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

### Matching Triangles

##### Stage: 1 Challenge Level:

Can you sort these triangles into three different families and explain how you did it?

### Inside Triangles

##### Stage: 1 Challenge Level:

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

### How Safe Are You?

##### Stage: 2 Challenge Level:

How much do you have to turn these dials by in order to unlock the safes?

### Six Places to Visit

##### Stage: 2 Challenge Level:

Can you describe the journey to each of the six places on these maps? How would you turn at each junction?

### Number the Sides

##### Stage: 2 Challenge Level:

The triangles in these sets are similar - can you work out the lengths of the sides which have question marks?

### Dotty Circle

##### Stage: 2 Challenge Level:

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

### Round and Round and Round

##### Stage: 3 Challenge Level:

Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?

### Can You Explain Why?

##### Stage: 3 Challenge Level:

Can you explain why it is impossible to construct this triangle?

### Where Is the Dot?

##### Stage: 3 Challenge Level:

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

### Sine and Cosine

##### Stage: 4 Challenge Level:

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

### Eight Ratios

##### Stage: 4 Challenge Level:

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

### Two Regular Polygons

##### Stage: 4 Challenge Level:

Two polygons fit together so that the exterior angle at each end of their shared side is 81 degrees. If both shapes now have to be regular could the angle still be 81 degrees?

### Wedge on Wedge

##### Stage: 4 Challenge Level:

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

### Over the Pole

##### Stage: 5 Challenge Level:

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.

### Flight Path

##### Stage: 5 Challenge Level:

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

### Pythagoras on a Sphere

##### Stage: 5 Challenge Level:

Prove Pythagoras' Theorem for right-angled spherical triangles.