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The points P, Q, R and S are the midpoints of the edges of a convex quadrilateral. What do you notice about the quadrilateral PQRS as the convex quadrilateral changes?

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Matter of Scale

Prove Pythagoras' Theorem using enlargements and scale factors.

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Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?

Flower Show

Stage: 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.