IMC 2014 Q16
16. The base of a triangle is increased by 25% but the area of the triangle is unchanged.
By what percentage is the corresponding perpendicular height decreased?
- A \(12\tfrac{1}{2}\)%
- B 16%
- C 20%
- D 25%
- E 50%
Intermediate Mathematical Challenge 2014, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 20%
If the areas of the original and new triangles are the same then the product of the base and the perpendicular height must be the same for the two triangles. When the base of the original triangle is increased by 25%, its value is multiplied by \(\tfrac{5}{4}\). So if the area is to remain unchanged then the perpendicular height must be multiplied by \(\tfrac{4}{5}\), which means that its new value is 80% of its previous value. So it is decreased by 20%.
Solution © UK Mathematics Trust, from the IMC 2014 solutions. Longer versions with investigations: Solutions and investigations (PDF).