Question

show that the sum of the three altitudes of a triangle is less than the sum of the three sides of the triangle

Answer 1

The shortest distance from a point to a line is the perpendicular. Hence each altitude is shorter than both of the triangles sides that that are incident at the angle the altitude is from. I.e. Altitude from A is shorter than both b and c, altitude from B is shorter than both both a and c, the altitude from C is shorter than both a and b. Now adding selectively the result follows.

Answer 2

Let there be a triangle ABC with its altitudes D, E, and F from vertices A, B and C respectively.

The altitudes form a right angle at their corresponding bases. Also, in a right triangle the hypotenuse is the longest side. Taking the right triangles formed by the altitudes and the sides as the hypotenuse, we observe that in each triangle, the side forms the longest side, i.e,

In triangle ABD, AB is the longest side

In triangle ACF, AC is the longest side

In triangle CBE, BC is the longest side

So, adding all the three, we get that the perimeter of a triangle is greater than the sum of its three altitudes.

Hope the answer is helpful !!

Good Day !!