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Monday, 5 May 2014

Sunday Afternoon Maths XI Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

Light Work

Extension
If instead a window 3/4 of the size was required, how could this be done?

Semi Circle in a Triangle

Label the triangle as follows:
EC and BC are both tangents to the circle so FEA is a right angle and the lengths EC and BC are equal, so the length of EC is 5. Let r be the radius of the circle.
Using Pythagoras' Theorem in triangle FEA:
$$8^2+r^2=(12-r)^2$$ $$64+r^2=144-24r+r^2$$ $$24r=80$$ $$r=\frac{80}{24}=\frac{10}{3}$$
Extension
What is the radius of the circle whose diameter lies on the 5cm side and to which the 12cm and 13cm sides are tangents?
What is the radius of the circle whose diameter lies on the 13cm side and to which the 12cm and 5cm sides are tangents?
Additional observation
For each pair of semi circles, draw a straight line between the two points where the semi circles intersect. These lines all meet at a point.