IMC 2007 Q25
25. The diagram shows a semi-circle and an isosceles triangle which have equal areas. What is the value of \(\tan x\)° ?
- A 1
- B \(\dfrac{\sqrt{3}}{2}\)
- C \(\dfrac{\pi}{\sqrt{3}}\)
- D \(\dfrac{2}{\pi}\)
- E \(\dfrac{\pi}{2}\)
Intermediate Mathematical Challenge 2007, Question 25. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E \(\dfrac{\pi}{2}\)
Let the radius of the semicircle be \(r\) and let the perpendicular height of the isosceles triangle be \(h\). Then \(\tan x\)° = \(\tfrac{h}{r}\).
Now the area of the semicircle = \(\tfrac{1}{2}\pi r^2\), whilst the area of the triangle = \(\tfrac{1}{2} \times 2r \times h = rh\). So \(rh = \tfrac{1}{2}\pi r^2\), giving \(\tfrac{h}{r} = \tfrac{\pi}{2}\).
Solution © UK Mathematics Trust, from the IMC 2007 solutions.