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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

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Lap Times

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the track.

Doesn't Add Up

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Patrick from Woodbridge School offered the following insight to this problem, as did Chip from King's Ely School.

The difference arises because there is a difference in the line gradient between the trapezium and the triangle. The sloping line of the trapezium has a gradient of $\frac{2}{5} = 0.4$ , whereas the hypotenuse of the triangle has a gradient of $\frac{3}{8} = 0.375$.
This means that the lines made when you join the triangle and trapezium together are not straight, so there is a small amount of space, as shown on my diagram. This space adds up to the extra $1^2$cm.


An anonymous solver also wrote:

The area of the original square was $64$ square units and the area of the rectangle was $65$ square units. What appears to be a diagonal of the rectangle is not in fact a straight line and the extra unit of area comes from the long thin parallelogram in the middle of the rectangle.

You can confirm this by making a careful scale drawing for yourself on graph paper.

The long thin parallelogram occurs because the slopes of the trapezia are not the same as explained by Patrick and Chip: the slopes of the triangles. For each trapezium the slope is in the ratio of 2 up to 5 across and for each triangle the slope is in the ratio of 3 up to 8 across.


Finally, Susie from DEECD Victoria offered an example of other numbers that work (Do they have the same properties as the numbers used for the lengths in the problem?). She said:

Another set of dimensions are $4,7,11$ in the corresponding places of $3,5,8$.
The areas are $121$ square units compared to $126$ square units.