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Coordinate Geometry - Midpoint and Length of a Line Segment

Grade 9IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A coordinate plane (Cartesian plane) uses a pair of numerical coordinates (x, y) to uniquely determine the position of a point.

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The Midpoint is the central point of a line segment that is equidistant from both endpoints.

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The Length (or Distance) of a line segment is the straight-line measurement between two points, calculated using the Pythagorean Theorem.

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When calculating distance, the order of points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) does not matter because the differences are squared.

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Horizontal lines have the same y-coordinates, while vertical lines have the same x-coordinates.

📐Formulae

Midpoint M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Length d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

💡Examples

Problem 1:

Find the midpoint of the line segment joining the points A(2,−3)A(2, -3) and B(8,7)B(8, 7).

Solution:

M=(2+82,−3+72)=(102,42)=(5,2)M = \left( \frac{2 + 8}{2}, \frac{-3 + 7}{2} \right) = \left( \frac{10}{2}, \frac{4}{2} \right) = (5, 2)

Explanation:

To find the midpoint, add the x-coordinates together and divide by 2, then add the y-coordinates together and divide by 2.

Problem 2:

Calculate the length of the line segment with endpoints P(1,2)P(1, 2) and Q(5,5)Q(5, 5).

Solution:

d=(5−1)2+(5−2)2=42+32=16+9=25=5d = \sqrt{(5 - 1)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 units

Explanation:

Substitute the coordinates into the distance formula. Subtract the x-values and y-values, square the results, sum them up, and finally take the square root.

Problem 3:

The midpoint of a line segment JKJK is M(4,5)M(4, 5). If point JJ is (1,2)(1, 2), find the coordinates of point KK.

Solution:

Let KK be (x,y)(x, y). 1+x2=4⇒1+x=8⇒x=7\frac{1 + x}{2} = 4 \Rightarrow 1 + x = 8 \Rightarrow x = 7. 2+y2=5⇒2+y=10⇒y=8\frac{2 + y}{2} = 5 \Rightarrow 2 + y = 10 \Rightarrow y = 8. Point KK is (7,8)(7, 8).

Explanation:

This is a 'reverse' midpoint problem. Set up two separate equations (one for x and one for y) using the midpoint formula and solve for the unknown coordinates of the endpoint.