Hamilton 2019 Q5
5. The area of the right-angled triangle in the diagram alongside is 60 cm2. The triangle touches the circle, and one side of the triangle has length 15 cm, as shown.
What is the radius of the circle?
Hamilton Olympiad 2019, Question 5. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: 20 cm
The area of a triangle is \(\tfrac{1}{2}\)base × height, so the triangle has sides of length 8 cm, 15 cm and 17 cm (using Pythagoras’ theorem).
Let the radius of the circle be \(r\) cm.
Draw two radii, as shown. Since a tangent meets the radius at the point of contact at right angles, we have a square, with sides of length \(r\) cm.
Then, using the equal tangent theorem, we have
so that \(2r = 40\) and \(r = 20\).
Commentary
The equal tangent theorem states that the two tangents from a point outside a circle to the circle are equal in length. Here we use two of the vertices of the triangle as the points.
Hence the radius of the circle is 20 cm.
Solution © UK Mathematics Trust, from the Hamilton Olympiad 2019 solutions.