Question

Segment MN with endpoints M(0, -3) and N(6,5) contains point K (3, 1). Determine what statement(s) explain how MK and NK are congruent or not congruent.

A. The segments are congruent because K is the midpoint.
B. The segments are congruent because the slopes are the same.
C. The segments are congruent because the distances are equal.
D. The segments are not congruent because the distances are not equal.
E. The segments are not congruent because the slopes are not the same.

Answer 1

Segments MK and NK are congruent because they have equal lengths, both measuring 5 units. The distance formula confirms this, making statement C the correct explanation for their congruence.

Step-by-step explanation:

To determine if segments MK and NK are congruent, we need to calculate their lengths and see if they are equal. We use the distance formula d = √((x2 - x1)^2 + (y2 - y1)^2) to find the lengths of MK and NK.

For segment MK, with endpoints M(0, -3) and K(3, 1), the length is √((3 - 0)^2 + (1 - (-3))^2) = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

For segment NK, with endpoints N(6, 5) and K(3, 1), the length is √((3 - 6)^2 + (1 - 5)^2) = √((-3)^2 + (-4)^2) = √(9 + 16) = √25 = 5.

Since both MK and NK have the equal length of 5, the segments are congruent. Therefore, the correct statement that explains why MK and NK are congruent is:

C. The segments are congruent because the distances are equal.