Question

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

Answer 1

you have to do a activity for it. take a rectangular strip of paper and make it a triangle ABC by folding its sides. now take another strip DE equal to side AB now try to fit it inside the triangle without folding it from anywhere. you'll find that it can't be fitted without folding. it means that that strip should be smaller so that it can be fitted. it means that the altitude is always smaller. repeat it with other sides and you'll get the same answer. now it means that every altitude is smaller than sides. hence sum of all sides of triangle is always bigger than sum of its altitudes.

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 !!