Question 1: Calculate the midpoint of the segment joining A(2, 4) and B(6, 8).

Answer: 4, 6

Question 2: Point C has coordinates (x, y). If the midpoint of AC is (5, 7) and A is at (3, 5), find the coordinates of C.

Answer: 7, 9

Question 3: A student calculates the midpoint of (1, 2) and (3, 4) as (2, 3). Identify and explain the mistake made.

Answer: Incorrectly averaged the x-coordinates as (1+3)/2=2 and y-coordinates as (2+4)/2=3, which is correct. The mistake might be in misreading coordinates or misapplication in a different context.

Question 4: Construct a line segment with endpoints at (−2, 3) and (4, −1). Mark the midpoint clearly on the grid.

Answer: -1, 1

Question 5: The coordinates of points P and Q are (x₁, y₁) and (x₂, y₂). The midpoint R is at (0, 0). If P is at (−6, 4), find Q.

Answer: 6, -4

Question 6: Explain why the midpoint formula is essential in proving that two segments are equal in length.

Answer: Because the midpoint formula finds the exact middle point, it helps verify if two segments are bisected equally, which implies they are of equal length if their endpoints are symmetric.

Question 7: A midpoint M of segment AB is at (3, 5). If A is at (1, 2), find B.

Answer: 5, 8

Question 8: Identify which of the following is an incorrect application of the midpoint formula: A) Averaging x-coordinates and y-coordinates B) Summing coordinates without dividing C) Using the formula for a midpoint D) Both A and B

Answer: D

Question 9: Given points D(−1, 4) and E(3, 10), what is the midpoint? Show your working.

Answer: 1, 7

Question 10: A line segment has endpoints at (2, −3) and (−2, 5). The midpoint is found to be (0, 1). Is this correct? Justify your answer.

Answer: Yes, because (2 + (-2))/2=0 and (−3 + 5)/2=1, which matches the given midpoint.

Question 11: Challenge question: Prove that the midpoint of a segment remains unchanged when the segment is translated parallel to itself.

Answer: Translation moves both endpoints equally, so their sum of coordinates remains the same relative to the midpoint formula, thereby keeping the midpoint unchanged.