Given:
[tex]$\begin{array}{c}
A B=12 \
A C=6
\end{array}$[/tex]
Prove: C is the midpoint of [tex]$\overline{ AB }$[/tex].
Proof:
We are given that [tex]$AB =12$[/tex] and [tex]$AC =6$[/tex]. Applying the segment addition property, we get [tex]$A C+C B=A B$[/tex].
Applying the substitution property, we get [tex]$6+ CB =12$[/tex].
The subtraction property can be used to find [tex]$CB =6$[/tex].
The symmetric property shows that [tex]$6=A C$[/tex]. Since [tex]$C B$[/tex] [tex]$=6$[/tex] and [tex]$6= AC , AC = CB$[/tex] by the property. So, [tex]$\overline{ AC } \cong \overline{ CB }$[/tex] by the definition of congruent segments. Finally, C is the midpoint of [tex]$\overline{ AB }$[/tex] because it divides [tex]$\overline{ AB }$[/tex] into two congruent segments.
Answer (2)
Use segment addition postulate: A C + CB = A B .
Substitute given values: 6 + CB = 12 .
Solve for CB : CB = 6 .
Conclude that C is the midpoint of A B since A C = CB = 6 .
Explanation
Problem Analysis We are given that segment AB has a length of 12 units, and segment AC has a length of 6 units. Our goal is to prove that point C is the midpoint of segment AB. To do this, we need to show that C divides AB into two equal segments, meaning AC must be equal to CB.
Apply Segment Addition Postulate We'll use the segment addition postulate, which states that if C is a point on segment AB, then AC + CB = AB. We know AB = 12 and AC = 6. Substituting these values into the equation, we get:
6 + CB = 12
Solve for CB To find the length of CB, we subtract 6 from both sides of the equation:
CB = 12 - 6
CB = 6
Compare AC and CB Now we know that AC = 6 and CB = 6. Since AC and CB have the same length, we can say that AC = CB. This means that point C divides segment AB into two congruent segments.
Conclusion By the definition of a midpoint, if a point divides a segment into two congruent segments, then that point is the midpoint of the segment. Since AC = CB, point C is the midpoint of segment AB.
Examples
In architecture, when designing a bridge, the midpoint of the bridge's span is crucial for placing support structures to ensure balanced weight distribution. Similarly, in sports, the midpoint of a track or field determines fair starting positions or turning points for athletes.
By applying the segment addition postulate and solving for the length of segment CB , we found that A C = CB = 6 . Since both segments are equal, point C is the midpoint of segment A B . Therefore, C divides AB into two congruent segments, confirming that C is the midpoint.
;
