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E3.4 Length and midpoint

Syllabus
0580–2028–2029
Topic
E3.4
Level
Extended

Calculate the length between two points

The horizontal and vertical changes between two points form the perpendicular sides of a right-angled triangle. The line segment is its hypotenuse, so its length follows from Pythagoras’ theorem.

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

Subtract corresponding coordinates to find Δx\Delta x and Δy\Delta y. Keep brackets around negative values, square both changes, add, then take the positive square root. The subtraction order does not affect the length because each difference is squared.

A(−5,2), B(7,8):d=(7−(−5))2+(8−2)2=180=65≈13.4A(-5,2),\ B(7,8):\quad d=\sqrt{(7-(-5))^2+(8-2)^2}=\sqrt{180}=6\sqrt5\approx13.4

The answer must be at least as large as the greater of ∣Δx∣|\Delta x| and ∣Δy∣|\Delta y|. Here 13.4>1213.4>12, so the result is plausible. If the points share an xx- or yy-coordinate, the formula reduces to the absolute difference on the other axis.

Do not add coordinate differences directly or round intermediate values. Square each complete signed difference and round only the final decimal when the required accuracy is known.

Find the midpoint of a line segment

The midpoint lies exactly halfway between the endpoints in both horizontal and vertical directions. Find it by averaging the two xx-coordinates and averaging the two yy-coordinates separately.

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

Pair like coordinates: add xx to xx and divide by 22, then add yy to yy and divide by 22. Use brackets when a coordinate is negative so the sign remains part of the sum.

P(−1,3), Q(6,4):M=(−1+62,3+42)=(2.5,3.5)P(-1,3),\ Q(6,4):\quad M=\left(\frac{-1+6}{2},\frac{3+4}{2}\right)=(2.5,3.5)

From PP to MM, the change is (3.5,0.5)(3.5,0.5); from MM to QQ, it is also (3.5,0.5)(3.5,0.5). Equal coordinate changes confirm that MM is halfway along the segment.

Do not divide only one coordinate or average an xx-coordinate with a yy-coordinate. A midpoint can have halves or other decimals even when both endpoints have integer coordinates.