2026 · Q87
For the next two (02) items that follow : ABC is an equilateral triangle. BP is perpendicular to AC, PR is perpendicular to AB, and PQ is perpendicular to BC. What is the ratio of AB² : BP² : PR²?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A4 : 3 : 1
- B16 : 12 : 3
- C16 : 12 : 5
- D5 : 4 : 3
Explanation
Take side s. BP is the altitude: BP² = s² − (s/2)² = 3s²/4. In right triangle APB, PR ⊥ AB with AP = s/2 gives PR = AP·sin 60° = (s/2)(√3/2) = s√3/4, so PR² = 3s²/16. Hence AB² : BP² : PR² = s² : 3s²/4 : 3s²/16 = 16 : 12 : 3, option B.