Question

given: m is the midpoint of $overline{ab}$. prove: a. $ab = 2cdot am$ b. $am=\frac{1}{2}ab$

Explanation:

Step1: Define mid - point property

By the definition of a mid - point, if \(M\) is the mid - point of \(\overline{AB}\), then \(AM = MB\) and \(AB=AM + MB\).

Step2: Substitute \(MB\) with \(AM\) in \(AB = AM+MB\)

Since \(AM = MB\), we have \(AB=AM + AM=2\cdot AM\).

Step3: Solve \(AB = 2\cdot AM\) for \(AM\)

Dividing both sides of the equation \(AB = 2\cdot AM\) by 2, we get \(AM=\frac{1}{2}AB\).

Answer:

a. Since \(M\) is the mid - point of \(\overline{AB}\), \(AB = AM+MB\) and \(AM = MB\), so \(AB = 2\cdot AM\).
b. Starting from \(AB = 2\cdot AM\), dividing both sides by 2 gives \(AM=\frac{1}{2}AB\).