Question

exercise #2: in the diagram below all points shown are collinear. h is the mid - point of (overline{ei}), f is the mid - point of (overline{eh}), and g is the mid - point of (overline{fh}). if (fg = 5) in, then what is the length of (overline{gi})? another concept closely related to the midpoint of a segment is that of a segment bisector. segment bisector any line, segment, or ray that passes through the midpoint of a given segment is known as a segment bisector. all segments have only one midpoint but an infinite number of segment bisectors. exercise #3: in quadrilateral abcd shown, diagonal (overline{bd}) bisects diagonal (overline{ac}). which two line segments must be congruent based on the information given? explain.

Explanation:

Step1: Analyze segment relationships in Exercise #2

Since \(G\) is the mid - point of \(\overline{FH}\), then \(FG = GH\). Given \(FG = 5\) in, so \(GH=5\) in. Also, \(H\) is the mid - point of \(\overline{EI}\), \(F\) is the mid - point of \(\overline{EH}\), but to find \(GI\), we know that \(GI=GH + HI\). Since \(H\) is the mid - point of \(\overline{EI}\), and we are not concerned with the other mid - points for finding \(GI\) from the given \(FG\). Since \(FG = GH = 5\) in, and \(GI=GH+HI\), and we know \(GH = 5\) in, and \(HI\) is not relevant as \(GI=GH+(HI)\) and we focus on the part related to \(G\) and \(H\). So \(GI = 15\) in.

Step2: Analyze Exercise #3

The definition of a segment bisector is that a line, segment, or ray that passes through the mid - point of a given segment. Since diagonal \(\overline{BD}\) bisects diagonal \(\overline{AC}\), by the definition of a segment bisector, the two congruent line segments are \(\overline{AE}\) and \(\overline{EC}\).

Answer:

Exercise #2: The length of \(\overline{GI}\) is \(15\) in.
Exercise #3: The congruent line segments are \(\overline{AE}\) and \(\overline{EC}\).