Question

additional geometric terminology n - gen math geometry
there is a great deal of terminology associated with the study of geometry. much of it uses descriptive names that help you understand the idea behind the term. all of this terminology must be mastered for this course. we begin with the concept of a mid - point
midpoint of a line segment
the midpoint is the unique point on a line segment that divides (or partitions) the line segment into two segments that are the same length (also known as two congruent segments).
exercise #1: given $overline{ab}$ shown, do the following:
(a) find the length of $overline{ab}$ in centimeters.
(b) based on (a), plot the midpoint of $overline{ab}$ and label it point $m$.
(c) make a formal statement about the congruence of two segments:
exercise #2: in the diagram below all points shown are collinear. $h$ is the midpoint of $overline{ei}$, $e$ is the midpoint of $overline{eh}$, and $g$ is the midpoint of $overline{fh}$. if $fg = 5$ in, then what is the length of $overline{gi}$?
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:

Exercise #1

(a)

Since no ruler - measurement or coordinate - based information is given for $\overline{AB}$, we cannot find its length numerically. Assume we have a ruler, we would place the zero - mark at point $A$ and read the value at point $B$ to get the length in centimeters.

(b)

If we knew the length of $\overline{AB}$ from part (a), say the length is $L$. Then the mid - point $M$ would be placed at a distance of $\frac{L}{2}$ from point $A$ along the line segment $\overline{AB}$.

(c)

The formal statement about the congruence of two segments is $\overline{AM}\cong\overline{MB}$ because the mid - point $M$ divides $\overline{AB}$ into two congruent segments.

Exercise #2

Step1: Identify the relationships based on mid - points

Since $G$ is the mid - point of $\overline{FH}$, then $\overline{FG}\cong\overline{GH}$ and $FH = 2FG$. Given $FG = 5$ in, so $FH=2\times5 = 10$ in.

Step2: Use the other mid - point relationships

Since $E$ is the mid - point of $\overline{EH}$ and $H$ is the mid - point of $\overline{EI}$, we know that $EI = 2EH$. Also, $FH$ is part of $EI$. The length of $\overline{GI}=FH = 10$ in.

Exercise #3

Step1: Recall the definition of a bisector

Since diagonal $\overline{BD}$ bisects diagonal $\overline{AC}$, by the definition of a segment bisector, the point of intersection (let's call it $E$) is the mid - point of $\overline{AC}$.

Step2: State the congruent segments

The two line segments that must be congruent are $\overline{AE}\cong\overline{EC}$ because a segment bisector passes through the mid - point of the segment it bisects.

Answer:

Exercise #1

(a) Cannot be determined without measurement.

(b) Place at half - length of $\overline{AB}$ from $A$ (if length known).

(c) $\overline{AM}\cong\overline{MB}$

Exercise #2

$10$ in

Exercise #3

$\overline{AE}\cong\overline{EC}$