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Broad Topics > Angles, Polygons, and Geometrical Proof > Perpendicular lines

### At Right Angles

##### Stage: 4 Challenge Level:

Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?

### Perpendicular Lines

##### Stage: 4 Challenge Level:

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

### 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.

##### Stage: 5 Challenge Level:

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

### Orthogonal Circle

##### Stage: 5 Challenge Level:

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

### Dotty Relationship

##### Stage: 2 Challenge Level:

Can you draw perpendicular lines without using a protractor? Investigate how this is possible.

### Right Angle Challenge

##### Stage: 1 Challenge Level:

How many right angles can you make using two sticks?

### Walls

##### Stage: 5 Challenge Level:

Plane 1 contains points A, B and C and plane 2 contains points A and B. Find all the points on plane 2 such that the two planes are perpendicular.

##### Stage: 4 Challenge Level:

Find the area of the shaded region created by the two overlapping triangles in terms of a and b?

### The Dodecahedron Explained

##### Stage: 5

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

### What Are You Plotting?

##### Stage: 2 Challenge Level:

Investigate the positions of points which have particular x and y coordinates. What do you notice?

### Similarly So

##### Stage: 4 Challenge Level:

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

### Tricircle

##### Stage: 4 Challenge Level:

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .

### Right Time

##### Stage: 3 Challenge Level:

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?

### Shrink

##### Stage: 4 Challenge Level:

X is a moveable point on the hypotenuse, and P and Q are the feet of the perpendiculars from X to the sides of a right angled triangle. What position of X makes the length of PQ a minimum?