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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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The midpoint of a line segment is the exact center point, calculated by finding the average of the xx-coordinates and the average of the yy-coordinates of the endpoints. If the endpoints are (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the midpoint MM is (x1+x22,y1+y22)\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right).

A coordinate plane showing a line segment from (2,2) to (8,6) with a midpoint M at (5,4).
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The length of a line segment is the distance between its two endpoints. It is derived from Pythagoras' Theorem: a2+b2=c2a^2 + b^2 = c^2, where the horizontal distance is Δx=x2−x1\Delta x = x_2 - x_1 and the vertical distance is Δy=y2−y1\Delta y = y_2 - y_1.

A right-angled triangle on a coordinate plane illustrating the distance formula components.
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When calculating distances, always square the differences before adding them. Since squares of real numbers are always non-negative, the distance dd will always be a positive value (or zero if points coincide).

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To find an unknown endpoint when given the midpoint and one endpoint, use the midpoint formula in reverse. Let (x1,y1)(x_1, y_1) be the known point and (xM,yM)(x_M, y_M) be the midpoint. The missing point (x2,y2)(x_2, y_2) is found by x2=2xM−x1x_2 = 2x_M - x_1 and y2=2yM−y1y_2 = 2y_M - y_1.

📐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.

Problem 4:

Calculate the length of the line segment ABAB where AA is at (2,1)(2, 1) and BB is at (10,7)(10, 7).

A line segment from A(2,1) to B(10,7) on a coordinate plane.

Solution:

Δx=10−2=8\Delta x = 10 - 2 = 8 Δy=7−1=6\Delta y = 7 - 1 = 6 d=82+62d = \sqrt{8^2 + 6^2} d=64+36d = \sqrt{64 + 36} d=100d = \sqrt{100} d=10d = 10

Explanation:

We identify the coordinates (x1,y1)=(2,1)(x_1, y_1) = (2, 1) and (x2,y2)=(10,7)(x_2, y_2) = (10, 7). We calculate the horizontal change (8) and vertical change (6), then apply the distance formula which is effectively the square root of the sum of these squares.

Problem 5:

Find the coordinates of the midpoint of the line segment joining the points C(−4,5)C(-4, 5) and D(2,−3)D(2, -3).

A line segment from C(-4,5) to D(2,-3) showing the midpoint M at (-1,1).

Solution:

xM=−4+22=−22=−1x_M = \frac{-4 + 2}{2} = \frac{-2}{2} = -1 yM=5+(−3)2=22=1y_M = \frac{5 + (-3)}{2} = \frac{2}{2} = 1 Midpoint M=(−1,1)\text{Midpoint } M = (-1, 1)

Explanation:

To find the midpoint, we sum the xx-coordinates and divide by 2, then sum the yy-coordinates and divide by 2. This gives the average position of the two endpoints.