### Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

### Hypotenuse Lattice Points

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

### Beelines

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

##### Age 14 to 16Challenge Level

This printable worksheet may be useful: Ladder and Cube.

Why do this problem?

This problem provides an opportunity to practise Pythagoras Theorem and solving three-dimensional geometric problems in a real-life context.

While this problem can be approached from various methods using similar shapes, trigonometry, or equations and gradients of a straight line, it also develops algebraic fluency and reasoning.

The problem yields an equation which students may not be able to solve, so there is an opportunity to explore it using graphing or spreadsheet software.

Possible approach

Start by asking students to draw a diagram to visualise the problem, and ask whether there is more than one position the ladder could be in. Then ask them to consider what they know is always true in this situation. If you want to encourage a coordinate geometry approach, you might want to label the floor and the wall as the x and y axes.

Then, you could let students guess the height, and check if their guess ‘works’. They could do this beginning from any other quantity, such as the distance between the foot of the ladder and the cube. Let them work in groups as they try out different numbers and develop a set of conditions that must be met, using Pythagoras’ Theorem and similar triangles. Circulate as they work, first encouraging them to record their results systematically and then to generalise: what happens if you guess that the height is $h$ metres?

Bring the group together to share their ideas. Several different sets of equations and/or expressions may have arisen, containing fractions and square roots. You might want to use graphs to represent the expressions. For example, the ‘gradient’ of the ladder is $\frac{h}{\sqrt{16-h^2}}$ and the ‘gradient’ of the triangle above the ladder is $h-1.$ You could plot these two expressions, or any similar expressions the students have come up with, by hand or using graphical software. You could also use a spreadsheet package like Excel to find solutions.

Alternatively, if you want to use a purely algebraic approach, you could ask students to identify lengths that are unknown but useful to work out the height of the ladder. You might allow students to label diagrams with their own choice of letters and to write down equations explaining the relationships between their letters. Different choices of unknowns and conditions result in different systems of simultaneous equations which all combine to give quartic polynomials. The neatest solution we’ve found (recorded in the student solutions) comes from setting the height of the top of the ladder and the distance of the foot of the ladder from the wall as unknowns.

Key questions

What quantities are fixed and what quantities are variable in this problem?

As the ladder is sliding against the wall, what is always true?

Are there any equal angles in the diagram so that we could find similar triangles and establish relationships between lengths of unknow segments?

Possible support

You could begin with the same situation, but with the height of the top of the ladder given and ask students to find the length of the ladder. Repeat this for a few different heights, and then introduce the problem.

Recap of similar triangles.

Recap of Pythagoras’ Theorem.

Possible extension

How high is the top of the ladder above the ground if the ladder is L meters long (or a different number)?

What is the angle between the ladder and the ground?