IMC 2013 Q20
20. Jack's teacher asked him to draw a triangle of area 7cm2. Two sides are to be of length 6cm and 8cm. How many possibilities are there for the length of the third side of the triangle?
- A 1
- B 2
- C 3
- D 4
- E more than 4
Intermediate Mathematical Challenge 2013, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 2
Let the base \(AB\) of the triangle be the side of length 8 cm and let \(AC\) be the side of length 6 cm. So \(C\) must lie on the circle with centre \(A\) and radius 6 cm as shown. The area of the triangle is to be 7 cm2, so the perpendicular from \(C\) to \(AB\) (or to \(BA\) produced) must be of length \(\tfrac{7}{4}\) cm.
The diagram shows the four possible positions of \(C\). However, since\(\angle BAC_1 = \angle BAC_3\) and \(\angle BAC_2 = \angle BAC_4\), these correspond to exactly two possibilities for the length of the third side \(AC\). The diagrams below show the two possibilities.
Solution © UK Mathematics Trust, from the IMC 2013 solutions. Longer versions with investigations: Solutions and investigations (PDF).