Prove that the sum of the lengths of the three altitudes of a triangle is less than the sum of the lengths of the three sides of the triangle.
Answer:
- Let AP, BQ, and CR be the altitudes of △ABC.
- We know that the perpendicular is the shortest line segment that can be drawn from a point outside the line to that line.
Thus, we have Adding the three equations, we have - Thus, the sum of the lengths of the three altitudes of a triangle is less than the sum of the lengths of the three sides of the triangle.