A right triangle has legs measuring 9 cm and 12 cm. What is the length of the altitude to the hypotenuse?

Tue, 05 May 2026 15:55:21 +0000

**A right triangle has legs measuring 9 cm and 12 cm. What is the length of the altitude to the hypotenuse?** Why are so many exploring the geometry of a triangle with legs 9 and 12 centimeters? This classic right triangle has sparked quiet conversations across math-focused communities, especially as learners connect geometry to real-world applications—engineering, architecture, and design. The altitude to the hypotenuse, though less intuitive, reveals elegant proportional relationships that matter in both science and everyday problem-solving. Understanding it helps build foundational spatial reasoning skills in an age where visual literacy and practical math intuition are increasingly valued. A right triangle with legs measuring 9 cm and 12 cm features a hypotenuse calculated using the Pythagorean theorem: √(9² + 12²) = √(81 + 144) = √225 = 15 cm. This 9-12-15 triangle forms a familiar 3-4-5 pattern scaled by 3, reinforcing predictable geometric patterns. Yet the question remains: how long is the altitude drawn from the right angle to this 15 cm hypotenuse? The answer lies in a straightforward principle rooted in area equivalence—solving complexity by reframing perspective. Because area can be calculated two ways, the altitude emerges naturally:
Area = (leg₁ × leg₂) / 2 = (9 × 12) / 2 = 54 cm²
Area = (hypotenuse × altitude) / 2
So, (15 × h) / 2 = 54 → 15h = 108 → h = 108 / 15 = 7.2 cm.
This elegant solution avoids trigonometry, remaining accessible while highlighting core geometric relationships.