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]$6+C B=12$[/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)
Apply segment addition property: A C + CB = A B .
Substitute given values: 6 + CB = 12 .
Solve for CB : CB = 6 .
Conclude that A C = CB , thus C is the midpoint of A B . 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 congruent segments, meaning A C = CB .
Apply Segment Addition Property The segment addition property states that if C is a point on 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
Use Subtraction Property To find the length of segment CB , we can use the subtraction property of equality. Subtract 6 from both sides of the equation:
6 + CB − 6 = 12 − 6
CB = 6
Compare Segment Lengths Now we know that A C = 6 and CB = 6 . Therefore, A C = CB .
Apply Definition of Congruent Segments The definition of congruent segments states that if two segments have the same length, then they are congruent. Since A C = CB = 6 , we can say that segment A C is congruent to segment CB , which is written as:
A C ≅ CB
Apply Definition of Midpoint The definition of a midpoint states that if a point divides a segment into two congruent segments, then that point is the midpoint of the segment. Since C divides segment A B into two congruent segments ( A C ≅ CB ), we can conclude that C is the midpoint of segment A B .
Conclusion Therefore, C is the midpoint of A B .
Examples
In architecture, understanding midpoints is crucial for symmetrical designs. For instance, when designing a bridge, the central support often needs to be placed at the midpoint to ensure equal distribution of weight and stability. Similarly, in interior design, placing a centerpiece at the midpoint of a table or a room creates a balanced and visually appealing aesthetic.
To show that point C is the midpoint of segment AB, we used the segment addition property to find that both segments AC and CB are equal to 6 units. Since both segments are congruent, it follows that C is the midpoint of AB. Thus, we conclude that C is the midpoint of A B .
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