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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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Pericut

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?

Weekly Problem 24 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

Solution

From the symmetry of the figure, the two circles must be concentric. Let their centre be $O$. Let the radius of the semicircles be $r$. Then the radius of the outer circle is $2r$ and, by Pythagoras' Theorem, the radius of the inner shaded circle is $\sqrt{r^2+r^2}$, that is $\sqrt{2}r$.

So the radii of the two circles are in the ratio $\sqrt{2}:2$, that is $1:\sqrt{2}$, and hence the ratio of their areas is $1:2$.

This problem is taken from the UKMT Mathematical Challenges.

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