Question

If a point C lies between two points A and B such that AC = BC, then prove that AC = \(\frac{1}{2}\) AB. Explain by drawing the figure.

Answer 1

We are given that,

AC = BC 

Also, point C lies between points A and B.

Adding AC to both sides:

\(AC + AC = BC + AC\) ... (1)

Here, (BC + AC) coincides with AB. This is based on the property that things which coincide with one another are equal to each other. Therefore, we have:

\(BC + AC = AB\) ... (2)

We also know the property that things which are equal to the same thing are equal to each other. Hence, from equations (1) and (2), we can deduce the following:

\(AC + AC = AB\)

This simplifies to:

\(2AC = AB\) ... (3)

Finally, dividing both sides of equation (3) by 2, we get:

\(AC = \frac{1}{2} AB\)

This concludes the proof that the length of AC is half the length of AB.