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King's Maths School

Challenge 447: Two semicircles

A gentle pair of geometry problems to start the new year!

(a) A triangle ABC has sides AB = 4, AC = 3, BC = 5. A semicircle is drawn so that its diameter lies on AB and one end of the diameter is at A. What is the largest possible radius of the semicircle if it fits inside the triangle?

(b) In the same triangle ABC, a semicircle is instead drawn so that its diameter lies on BC. What is the largest possible radius of the semicircle if it fits inside the triangle?

Submit your solution

Please do send in your solution to this problem to weeklymaths@kcl.ac.uk You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.