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.
Solution and Explanation
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.
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