Question

If in the following figure, it is given that ED = EC, then prove that AB + AD > BC.

Answer 1

Hint: We know that the sum of two sides of a triangle is always greater than the third side. We must use this property in triangle ABD and then substitute ED with EC. Now, we will once again need to use the property in triangle BEC to get the desired result easily.

Complete step by step answer:
We know that in a triangle, the sum of any two sides is always greater than the third side.
So, for any triangle ABC, we have

AB + AC > BC
AB + BC > AC
AC + BC > AB

In our problem, we have the following figure,
We can use the above triangle inequality in triangle ABD. So, we get
AB + AD > BD…(i)
We can clearly see that E is a point on the line BD.
Thus, we have BD = BE + ED.
Putting the value of BD in equation (i), we get
AB + AD > BE + ED
We are given in the question that ED = EC.
So, we can also write,
AB + AD > BE + EC…(ii)
Now in triangle BEC, using the triangle inequality, we can write
BE + EC > BC…(iii)
From equation (ii) and equation (iii), we can write
AB + AD > BE + EC > BC
Hence, we can conclude that
AB + AD > BC

Note: We must be careful while selecting a triangle inequality out of three. We must select in accordance with our question and the needs of the problem. We must remember that the sum of two sides is always greater than the third side, and the difference of two sides is always less than the third side.