Question

Show that the sum of the three altitudes of a triangle is less than the sum of its three sides.

Answer 1

Hint: Construct the altitudes of the triangles and consider the smaller triangles as a result of the construction. Use the fact that the sides opposite to greater angles are longer than the sides opposite to smaller angles, to prove that the sum of the three altitudes of a triangle is less than the sum of its three sides.

Complete Step-by-Step solution:

Consider a triangle ABC and let AF, BE, and CD be the three altitudes of the triangle from the vertex A, B, and C respectively.
 The sides opposite to greater angles are longer than the sides opposite to the smaller angles.
Consider the triangle AFC, we have angle AFC equal to 90°, and angle C is an acute angle.
\[\angle AFC = 90^\circ \]
\[\angle C < 90^\circ \]
Hence, the angle AFC is greater than angle C.
\[\angle AFC > \angle C\]
The side opposite to angle AFC is AC and the side opposite to angle C is AF. Then, by the above-mentioned property, we have:
\[AF < AC.............(1)\]
Consider the triangle BAE, we have angle BEA equal to 90° and angle A is an acute angle.
\[\angle BEA = 90^\circ \]
\[\angle A < 90^\circ \]
Hence, the angle BEA is greater than angle A.
\[\angle BEA > \angle A\]
The side opposite to angle BEA is AB and the side opposite to angle A is BE. Then, by the above-mentioned property, we have:
\[BE < AB.............(2)\]
Consider the triangle BDC, we have angle BDC equal to 90°, and angle B is an acute angle.
\[\angle BDC = 90^\circ \]
\[\angle B < 90^\circ \]
Hence, the angle BDC is greater than angle B.
\[\angle BDC > \angle B\]
The side opposite to angle BDC is BC and the side opposite to angle B is CD. Then, by the above-mentioned property, we have:
\[CD < BC.............(3)\]
Adding equations (1), (2), and (3), we have:
\[AF + BE + CD < AC + AB + BC\]
Hence, the sum of the three altitudes of a triangle is less than the sum of its three sides.

Note: You can also use the property that the sum of two sides of the triangle is greater than the third side to prove that the sum of the three altitudes of a triangle is less than the sum of its three sides.