Question

Two sides of a triangle are 5cm and 9cm long. What can be the length of its third side?
AM is a median of a triangle ABC. Is AB + BC + CA > 2M? (consider the sides of triangles AABM
and JAMC).​

Answer 1

A(0,0) and B(9,0) are two points on x-axis and AB=9cm here.

This side we have to take constant.

For the 5cm side let's make a circle having centre at B and radius 5 cm.

Equation of the circle is (x−9)2+y2=25.

Now the circle crosses the x-axis at the point P(4,0) and Q(14,0). See it on the graph.

Now AB is a stationary side and any variable point C on the circle from B can make BC=5cm both above +x axis and below +x-axis. C can be varied from point P to point Q in both above +x-axis and below +x-axis.

C point can be varied from P(4,0) to Q(14,0)nothing more or less than that because the circle is of 5cm only.

Now the question asks us about triangle. If I will join ABP or ABQ then the three points will be colinear(lying on the same line),so according to basic rule of triangle ABP and ABQ can not be a triangle.

Except these two points P and Q on the circle other points can give the third side AC.

So you can see AC≠4cm,AC≠14cm

So the third side should be any value between 4cm and 14cm.