Question

determine whether the given conjecture is true or false. give a counterexample for any false conjecture. if s, t, and u are collinear and st = tu, then t is the midpoint of $overline{su}$.

Explanation:

Step1: Recall the definition of mid - point

The mid - point of a line segment divides the segment into two equal parts. In the case of collinear points \(S\), \(T\), and \(U\), if \(T\) is the mid - point of \(\overline{SU}\), then \(ST = TU\) and \(S\), \(T\), \(U\) are arranged such that \(S - T - U\) (in that order).

Step2: Consider the order of points

The given conjecture says if \(S\), \(T\), and \(U\) are collinear and \(ST = TU\), then \(T\) is the mid - point of \(\overline{SU}\). But we can have a situation where the order of the points is \(T - S - U\) or \(U - S - T\). For example, if the points are arranged in the order \(T - S - U\) and \(ST=TU\), \(T\) is not the mid - point of \(\overline{SU}\).

Answer:

False. Counter - example: Consider the collinear points arranged as \(T - S - U\) such that the distance \(ST = TU\). In this case, \(T\) is not the mid - point of \(\overline{SU}\).