Each of the two equal sides of an isosceles triangle is 18 cm long and the angle between them is 90°. What is the length of the shortest altitude of the triangle?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(9\sqrt{2}\) cm
An altitude is shortest when it falls on the longest side, because twice the area is the same for every side: side × its altitude. In a right triangle the longest side is the hypotenuse.
The two equal sides are at right angles, so each is the altitude to the other: both these altitudes are 18 cm
Hypotenuse \(=\sqrt{18^2+18^2}=18\sqrt{2}\) cm
Area \(\displaystyle =\frac{1}{2}\times18\times18=162\) cm2
Altitude to the hypotenuse \(\displaystyle =\frac{2\times162}{18\sqrt{2}}=\frac{18}{\sqrt{2}}=9\sqrt{2}\approx12.7\) cm
\(9\sqrt{2}\) cm is less than 18 cm, so it is the shortest altitude
Why the other options are incorrect
Option (1) 18 cm — Incorrect. This is the altitude to each equal side, the longest altitude of this triangle.
Option (2) 9 cm — Incorrect. This is half of a leg; no altitude of the triangle has this length.
Option (4) \(18\sqrt{2}\) cm — Incorrect. This is the length of the hypotenuse itself, not an altitude.