Question

to prove that c divides $overline{ab}$ into two congruent segments, we need to show that c is the midpoint of $overline{ab}$ using the substituting the coordinates of a and b into the midpoint formula, we will have performing the operation, we will get the midpoint, we have shown that the midpoint of $overline{ab}$ is the same as point c. therefore, by the definition of a midpoint, we have proven that c divides $overline{ab}$ into two congruent segments

Explanation:

Step1: Recall mid - point concept

To prove $C$ divides $\overline{AB}$ into two congruent segments, we use the mid - point definition.

Step2: State mid - point formula

If $A=(x_1,y_1)$ and $B=(x_2,y_2)$, the mid - point formula is $M = (\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step3: Substitute coordinates

Substitute the coordinates of $A$ and $B$ into the formula: $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step4: Calculate mid - point

Perform the arithmetic operations to get the mid - point coordinates.

Step5: Compare with point C

If the calculated mid - point is the same as point $C$, then $C$ is the mid - point of $\overline{AB}$ and divides it into two congruent segments.

Answer:

By showing that $C$ is the mid - point of $\overline{AB}$ using the mid - point formula, we prove that $C$ divides $\overline{AB}$ into two congruent segments.