Use the connection you found to find the coordinates of B given that M (−7, 1) is the midpoint of A (3, −4) and B (x, y).
Step-by-step solution
To find: B
Idea: The midpoint’s coordinates are the averages of the end points’ coordinates: M = (x1 + x22, y1 + y22). Knowing M and one end, solve for the other end.
- x-coordinates: 3 + x2 = −7, so 3 + x = −14 and x = −17.½ mark
- y-coordinates: −4 + y2 = 1, so −4 + y = 2 and y = 6.½ mark
- So B = (−17, 6).½ mark
- Check: midpoint of A (3, −4) and B (−17, 6) = ((3 − 17) ÷ 2, (−4 + 6) ÷ 2) = (−7, 1) = M ✓.½ mark
Check: Going from A to M is a shift of (−10, +5); going the same shift again from M gives (−7 − 10, 1 + 5) = (−17, 6) = B ✓.
Answer to write in the exam
M = (x1 + x22, y1 + y22)
3 + x2 = −7 ⇒ 3 + x = −14 ⇒ x = −17
−4 + y2 = 1 ⇒ −4 + y = 2 ⇒ y = 6
∴ B = (−17, 6)
Common mistakes that cost marks
- Averaging A and M instead: ((3 − 7) ÷ 2, (−4 + 1) ÷ 2) = (−2, −1.5). M is the midpoint, so solve for B.
- Forgetting to multiply by 2: writing 3 + x = −7, so x = −10.
- Sign slip: −4 + y = 2 gives y = 6, not −2.
How this can come in the exam
If (2, 3) is the midpoint of the segment joining (−1, y) and (x, 5), then (x, y) =
- (5, 1)
- (1, 5)
- (3, −2)
- (5, −1)
Show answer
(A) (5, 1)
(−1 + x) ÷ 2 = 2 gives x = 5; (y + 5) ÷ 2 = 3 gives y = 1.
The centre of a circle is (2, −3) and one end of a diameter is (−4, 5). Find the other end of the diameter.
Show answer
The centre is the midpoint of the diameter (½ mark). (−4 + x) ÷ 2 = 2 ⇒ x = 8; (5 + y) ÷ 2 = −3 ⇒ y = −11 (1 mark). Other end (8, −11) (½ mark).Try one yourself
M (0, 4) is the midpoint of A (−6, 9) and B. Find B.
Show answer
(−6 + x) ÷ 2 = 0 ⇒ x = 6; (9 + y) ÷ 2 = 4 ⇒ y = −1. B = (6, −1).
More questions like this
- Let P, Q be points of trisection of AB, with P closer to A, and Q closer to B. Using your knowledge of how to find the coordinates of the midpoint of a segment, how would you find the coordinates of P and Q? Do this for the case when the points are A (4, 7) and B (16, −2).
- (i) Given the points A (1, −8), B (−4, 7) and C (−7, −4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?
(ii) Given the points D (−5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K. - The midpoints of the sides of triangle ABC are the points D, E, and F. Given that the coordinates of D, E, and F are (5, 1), (6, 5), and (0, 3), respectively, find the coordinates of A, B and C.
- A city has two main roads which cross each other at the centre of the city. These two roads are along the North–South (N–S) direction and East–West (E–W) direction. All the other streets of the city run parallel to these roads and are 200 m apart. There are 10 streets in each direction.
- A computer graphics program displays images on a rectangular screen whose coordinate system has the origin at the bottom-left corner. The screen is 800 pixels wide and 600 pixels high. A circular icon of radius 80 pixels is drawn with its centre at the point A (100, 150). Another circular icon of radius 100 pixels is drawn with its centre at the point B (250, 230). Determine: