EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Line segments
Graphs · Lesson 5 of 8
Last Lesson and Before
Answer each one, then check.
1. What is the mean of 2 and 8?
5
2. Distance between (1, 1) and (4, 5)?
5
3. Work out (−3 + 7)/2.
2
4. (Higher) Gradient perpendicular to 2?
−½
Learning Objectives
1. Find the midpoint of a line segment.
2. Find the length of a line segment.
3. Find a missing end point from the midpoint.
4. (Higher) Find the equation of the perpendicular bisector of a line segment.
Midpoints
A line segment is the part of a line between two end points. Its midpoint is exactly halfway along.
▸ The mean of the ends. Midpoint of (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).
▸ Example. Midpoint of (2, 3) and (8, 11) is (10/2, 14/2) = (5, 7).
▸ Negatives. Take care adding negatives: midpoint of (−3, 5) and (7, 1) is (2, 3).
▸ Length. Use Pythagoras with the horizontal and vertical differences.
Midpoint and Length
|
To get from A to B you go 6 across and 8 up. The midpoint is half of each: 3 across and 4 up from A, which is (5, 7). The length of AB is √(6² + 8²) = √100 = 10. |
M is halfway across and halfway up: (5, 7). The length is √(6² + 8²) = 10. |
Finding the Other End
|
M(3, 1) is the midpoint of the line segment AB. A is (−1, 4). Find the coordinates of B. |
1. From A to M
x: −1 to 3 is +4; y: 4 to 1 is −3
2. Do the same again from M to B
x: 3 + 4 = 7; y: 1 − 3 = −2
3. Check the midpoint of A and B
((−1 + 7)/2, (4 + (−2))/2) = (3, 1)
Answer: B(7, −2)
PART TWO · HIGHER
Perpendicular Bisectors
The line that cuts a segment in half at right angles.
The Perpendicular Bisector
It passes through the midpoint, and is perpendicular to the segment.
|
1 Midpoint Find the midpoint of the segment. |
2 Gradient of the segment (change in y)/(change in x). |
3 Perpendicular gradient Take the negative reciprocal. |
4 Find c Substitute the midpoint into y = mx + c. |
A Perpendicular Bisector
|
A is (1, 2) and B is (5, 10). Find the equation of the perpendicular bisector of AB. |
1. Midpoint
((1 + 5)/2, (2 + 10)/2) = (3, 6)
2. Gradient of AB
(10 − 2)/(5 − 1) = 8/4 = 2
3. Perpendicular gradient
−½
4. Substitute (3, 6)
6 = −½ × 3 + c, so c = 7.5
Answer: y = −½x + 7.5, or x + 2y = 15
Key Terms
|
Line segment The part of a line between two end points. |
Midpoint The point exactly halfway along a line segment. |
|
Bisect Cut into two equal parts. |
Perpendicular bisector A line that cuts a segment in half at right angles. |
Your Task: Hidden Square
10 minutes
|
Three corners of a square are (1, 1), (5, 2) and (4, 6). (a) Find the midpoint of each diagonal you can draw. (b) Use the fact that the diagonals of a square bisect each other to find the fourth corner. (c) Find the length of one side. 1. Sketch the three points. 2. Decide which points are opposite corners. 3. Use the shared midpoint. |
A good answer shows: (a) The diagonal from (1, 1) to (4, 6) has midpoint (2.5, 3.5). (b) The other diagonal has the same midpoint, so the fourth corner is (2 × 2.5 − 5, 2 × 3.5 − 2) = (0, 5). (c) √(4² + 1²) = √17 = 4.12.
Can I...?
☐ Find the midpoint of a line segment.
☐ Find the length of a line segment.
☐ Find an end point from the midpoint and the other end.
☐ (Higher) Find the gradient perpendicular to a segment.
☐ (Higher) Find the equation of a perpendicular bisector.
Summary
✓ Midpoint: the mean of the xs and the mean of the ys.
✓ Length: Pythagoras on the differences.
✓ Missing end: same step again from the midpoint.
✓ (Higher) Perpendicular bisector: midpoint plus negative reciprocal gradient.
|
EXAM FOCUS M(3, 1) is the midpoint of AB. A has coordinates (−1, 4). Find the coordinates of B. (2 marks) For a missing end point, don't halve anything: find the step from A to M, then take the same step again from M. |