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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

Flower Show

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2
$9$ hours.

$$ \frac{\text{time taken to dig a bed of diameter six metres}}{\text{time taken to dig a bed of diameter four metres}}=\frac{6^2}{4^2}=\frac{3^2}{2^2} = \frac{9}{4}$$
Hence the time taken for one gardener to dig a bed of diameter six metres is $ \frac{9}{4} \times4 \; \text{hours} = 9 \; \text{hours}$

The time taken for six gardeners to dig six flower beds is the same as the time it takes one gardener to dig one bed of the same size, and hence the answer is $9$ hours.

This problem is taken from the UKMT Mathematical Challenges.
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