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Midpoint of a segment · 5 marks

The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer.

  1. Row 1 S (−3, 0), M (0, 0), T (3, 0)
  2. Row 2 S (2, 3), M (3, 4), T (4, 5)
  3. Row 3 S (0, 0), M (0, 5), T (0, −10)
  4. Row 4 S (−8, 7), M (0, −2), T (6, −3)
  5. Connection When M is the mid-point of ST, can you find any connection between the coordinates of M, S and T?
Answer: Row 1: Yes (SM = MT = 3, ST = 6) · Row 2: Yes (SM = MT = √2, ST = 2√2) · Row 3: No (SM = 5, MT = 15) · Row 4: No (SM = √145, MT = √37). Connection: the midpoint’s coordinates are the averages, M = (xS + xT2, yS + yT2).

Step-by-step solution

Idea: M is the midpoint of ST when M lies on ST and is equally far from S and T, that is, SM = MT and SM + MT = ST. Checking the four rows this way reveals the pattern: each coordinate of the midpoint is halfway between the coordinates of S and T.

Row 1 S (−3, 0), M (0, 0), T (3, 0)

  1. All three points are on the x-axis. SM = 0 − (−3) = 3, MT = 3 − 0 = 3, ST = 3 − (−3) = 6.½ mark
  2. SM = MT and SM + MT = ST, so Yes, M is the midpoint.½ mark
Yes

Row 2 S (2, 3), M (3, 4), T (4, 5)

  1. SM = √(12 + 12) = √2, MT = √(12 + 12) = √2, ST = √(22 + 22) = √8 = 2√2.½ mark
  2. SM = MT and SM + MT = 2√2 = ST, so Yes, M is the midpoint.½ mark
Yes

Row 3 S (0, 0), M (0, 5), T (0, −10)

  1. All three points are on the y-axis. SM = 5 − 0 = 5, but MT = 5 − (−10) = 15.½ mark
  2. SM ≠ MT (in fact M is not even between S and T: S is between M and T). So No. The midpoint of ST is (0, −5).½ mark
No

Row 4 S (−8, 7), M (0, −2), T (6, −3)

  1. SM = √(82 + (−9)2) = √(64 + 81) = √145, MT = √(62 + (−1)2) = √37.½ mark
  2. SM ≠ MT, so No. (The midpoint of ST is (−1, 2).)½ mark
No

Connection When M is the mid-point of ST, can you find any connection between the coordinates of M, S and T?

  1. Row 1: (−3 + 3) ÷ 2 = 0 and (0 + 0) ÷ 2 = 0, which is M (0, 0). Row 2: (2 + 4) ÷ 2 = 3 and (3 + 5) ÷ 2 = 4, which is M (3, 4).½ mark
  2. So each coordinate of the midpoint is the average of the corresponding coordinates of S and T: for S (x1, y1) and T (x2, y2), M = (x1 + x22, y1 + y22). In rows 3 and 4 the averages, (0, −5) and (−1, 2), are not M, which agrees with the answers “No”.½ mark
M = (x1 + x22, y1 + y22): each coordinate of M is the average of those of S and T.
Row 1: Yes. Row 2: Yes. Row 3: No (SM = 5, MT = 15). Row 4: No (SM = √145, MT = √37). Connection: the midpoint of S (x₁, y₁) and T (x₂, y₂) is M ((x₁ + x₂)/2, (y₁ + y₂)/2).

Check: Use the connection on every row: (0, 0) ✓, (3, 4) ✓, (0, −5) ≠ (0, 5) ✓, (−1, 2) ≠ (0, −2) ✓. Same answers as the distance method.

Answer to write in the exam

Row 1

SM = 0 − (−3) = 3, MT = 3 − 0 = 3, ST = 6

SM = MT and SM + MT = ST

∴ Yes, M is the midpoint of ST.

Row 2

SM = √(1 + 1) = √2, MT = √(1 + 1) = √2

ST = √(4 + 4) = 2√2 = SM + MT

∴ Yes, M is the midpoint of ST.

Row 3

SM = 5, MT = 5 − (−10) = 15

SM ≠ MT

∴ No, M is not the midpoint of ST (midpoint is (0, −5)).

Row 4

SM = √(82 + 92) = √145, MT = √(62 + 12) = √37

SM ≠ MT

∴ No, M is not the midpoint of ST.

Connection

Row 1: ((−3 + 3)/2, (0 + 0)/2) = (0, 0) = M

Row 2: ((2 + 4)/2, (3 + 5)/2) = (3, 4) = M

∴ Midpoint of S (x1, y1) and T (x2, y2) is M (x1 + x22, y1 + y22).

Common mistakes that cost marks

  • Checking only SM = MT. A point can be equally far from S and T without being on ST; the midpoint must also lie on the segment (SM + MT = ST).
  • In row 3, taking MT = 5 − 10 = −5. With T (0, −10), MT = 5 − (−10) = 15.
  • Writing the connection as M = (x2 − x1)/2. The midpoint uses the sum of the coordinates, not the difference.

How this can come in the exam

MCQ (1 mark)

The midpoint of the segment joining (−4, 6) and (10, −2) is

  1. (3, 2)
  2. (7, 4)
  3. (6, 4)
  4. (3, −4)
Show answer

(A) (3, 2)
((−4 + 10) ÷ 2, (6 + (−2)) ÷ 2) = (3, 2).

Assertion–Reason (1 mark)

Assertion (A): (1, 1) is the midpoint of the segment joining (−2, 3) and (4, −1).
Reason (R): Each coordinate of the midpoint is the average of the corresponding coordinates of the end points.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
(−2 + 4) ÷ 2 = 1 and (3 + (−1)) ÷ 2 = 1, by R.

Try one yourself

Is M (2, −1) the midpoint of S (−1, 3) and T (5, −5)? Justify.

Show answer

Yes. SM = √(9 + 16) = 5 and MT = √(9 + 16) = 5, ST = √(36 + 64) = 10 = SM + MT. Also ((−1 + 5) ÷ 2, (3 − 5) ÷ 2) = (2, −1).

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← Using the origin as one vertex, plot the vertices of:Use the connection you found to find the coordinates of B given that M →