Given: [tex]AB =12[/tex]
[tex]A C=6[/tex]
Prove: C is the midpoint of [tex]\overline{ AB }[/tex].
Proof:
We are given that [tex]A B=12[/tex] and [tex]A C=6[/tex]. Applying the segment addition property, we get [tex]AC + CB = AB[/tex].
Applying the substitution property, we get [tex]6+C B=12[/tex].
The subtraction property can be used to find [tex]C B=6[/tex]. The symmetric property shows that [tex]6=A C[/tex]. Since [tex]C B=6[/tex]
and [tex]6=A C, A C=C B[/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)
Apply 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 . C is the midpoint of A B
Explanation
Problem Analysis We are given that segment A B has a length of 12, and segment A C has a length of 6. Our goal is to prove that point C is the midpoint of segment A B . To do this, we need to show that C divides A B into two equal segments, meaning A C = CB .
Apply Segment Addition Postulate The segment addition postulate states that if C is a point on the line segment A B , then A C + CB = A B . We know A B = 12 and A C = 6 , so we can substitute these values into the equation:
6 + CB = 12
Solve for CB To find the length of CB , we can subtract 6 from both sides of the equation:
6 + CB − 6 = 12 − 6
CB = 6
Compare AC and CB Now we know that A C = 6 and CB = 6 . Therefore, A C = CB . This means that point C divides the segment A B into two segments of equal length.
Conclusion By the definition of a midpoint, if a point divides a segment into two congruent segments, then it is the midpoint of the segment. Since A C = CB , we can conclude that C is the midpoint of A B .
Examples
In architecture, when designing a bridge, the midpoint of the bridge's span is crucial for evenly distributing weight and ensuring structural balance. Similarly, in sports, the midpoint of a track or field determines fair starting points for races. Understanding midpoints helps in creating balanced and symmetrical designs in various real-world applications.
Point C is the midpoint of segment A B because it divides the segment into two equal parts: both A C and CB measure 6 units. By applying the segment addition postulate and solving for the unknown segment length, we confirm that both lengths are equal. Therefore, C is the midpoint as defined in geometry.
;
