CIE IGCSE Mathematics (0580) Length and midpoint Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Length and midpoint Study Notes
LEARNING OBJECTIVE
- Understanding Length and midpoint of line segment.
Key Concepts:
- Length and midpoint
Length of a Line Segment
To find the distance (or length) between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), use the distance formula:
\( \text{Length} = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \)
This formula is derived from the Pythagorean Theorem by treating the line segment as the hypotenuse of a right-angled triangle.
Example:
Find the length of the line segment joining the points \( A(1, 2) \) and \( B(5, 5) \).
▶️ Answer/Explanation
Use the formula:
\( \text{Length} = \sqrt{(5 – 1)^2 + (5 – 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \)
Answer: 5 units
Example:
Find the length of the segment joining \( P(-2, -3) \) and \( Q(4, 1) \).
▶️ Answer/Explanation
Use the formula:
\( \text{Length} = \sqrt{(4 – (-2))^2 + (1 – (-3))^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} \)
\( \text{Length} = 2\sqrt{13} \approx 7.21 \ \text{units} \)
Answer: \( 2\sqrt{13} \) or approximately 7.21 units
Midpoint of a Line Segment
The midpoint of a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is the point exactly halfway between them.
$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $
This formula averages the x-coordinates and the y-coordinates of the endpoints.
Example:
Find the midpoint of the line segment joining the points \( A(2, 6) \) and \( B(8, 10) \).
▶️ Answer/Explanation
Use the midpoint formula:
$ \text{Midpoint} = \left( \frac{2 + 8}{2}, \frac{6 + 10}{2} \right) = \left( \frac{10}{2}, \frac{16}{2} \right) = (5, 8)$
Answer: (5, 8)
Example:
Find the midpoint of the segment joining \( P(-3, 4) \) and \( Q(5, -2) \).
▶️ Answer/Explanation
Use the midpoint formula:
$ \text{Midpoint} = \left( \frac{-3 + 5}{2}, \frac{4 + (-2)}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1) $
Answer: (1, 1)
Example:
The midpoint of a line segment is \( M(4, 3) \), and one endpoint is \( A(2, 1) \). Find the coordinates of the other endpoint \( B \), given that the length of the segment is \( 4\sqrt{2} \).
▶️ Answer/Explanation
$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \Rightarrow (4, 3) = \left( \frac{2 + x}{2}, \frac{1 + y}{2} \right) $
Equating components:
$ \frac{2 + x}{2} = 4 \Rightarrow 2 + x = 8 \Rightarrow x = 6 $
$ \frac{1 + y}{2} = 3 \Rightarrow 1 + y = 6 \Rightarrow y = 5 $
So the other endpoint is \( B(6, 5) \).
Confirm length using distance formula:
$ AB = \sqrt{(6 – 2)^2 + (5 – 1)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} $
Final Answer: \( B(6, 5) \)