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King's Maths School

Challenge 410: In what circle?

Two expressions for a radius lead to a neat proof of a familiar result!

Take a right angled triangle with sides a, b, c, where c is the hypotenuse.

The incircle is the largest circle that can be drawn inside the triangle, and is tangent to all three sides of the triangle.

Let the radius of the incircle be r.

(a) By writing two different expressions for the perimeter (one should include a, b, c; the other should include a, b, r) find an expression for r.

(b) By writing two different expressions for the area (one should include a, b; the other should include a, b, c, r) find another expression for r.

(c) If you've found the same two expressions that I did, you should be able to equate them and derive a neat proof of a familiar result!

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.