Calculating Distance and Midpoint Between Two Points in Space
Learn how to find the distance and midpoint between two points in space using the coordinates provided. Understand the concepts of horizontal and vertical distances, Pythagoras' theorem, and a general formula for finding distances between points. Worked examples demonstrate the step-by-step calculations for clarity.
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16 June 2025 Distance between two points and their midpoint LO: To calculate the distance between two points in the space and their midpoint. www.mathssupport.org www.mathssupport.org
The distance between two points What is the distance between the points A(-3, 2) and B(3, -2)? y D(2, 4) 4 X 3 C(-3, 2) The horizontal distance between the points is 3 ( 3) = 6 2 X 1 E(-2, 0) X 1 0 -5 -4 -3 -2 -1 2 4 5 3 x -1 -2 X X B(3, 2) A(-3, 2) -3 The horizontal distance between any pair of horizontal points isx2 x1 www.mathssupport.org www.mathssupport.org
The distance between two points What is the distance between the points A(-3, 2) and C( 3, 2)? y D(2, 4) 4 X 3 C(-3, 2) 2 X The vertical distance between the points is 2 (-2) = 4 1 E(-2, 0) X 1 0 -5 -4 -3 -2 -1 2 4 5 3 x -1 -2 X X B(3, 2) A(-3, 2) -3 The vertical distance between any pair of vertical points is . y2 y1 www.mathssupport.org www.mathssupport.org
The distance between two points What is the distance between the points C(-3, 2) and D(2, 4)? y We can find the distance between them by adding a third point, F, to form a right-angled triangle. We then use Pythagoras theorem. LengthCF= 2 (-3) X D(2, 4) 4 2 3 C(-3, 2) X F(2, 2) 2 X 5 1 E(-2, 0) X x 1 0 -5 -4 -3 -2 -1 2 4 5 3 -1 = 5 A(-3, 2) Length DF= 4 2 -2 X = 2 -3 CD2 = 52 + 22 CD = 29 Using Pythagoras theorem. www.mathssupport.org www.mathssupport.org
Generalization for the distance between two points What is the distance between two general points with coordinates A(x1, y1) and B(x2, y2)? The horizontal distance between the points is . The vertical distance between the points is . Using Pythagoras Theorem, the square of the distance between the points A(x1, y1) and B(x2, y2) is x2 x1 y2 y1 ???= ?? ?? ?+ ?? ?? ? The distance between the points A(x1, y1) and B(x2, y2) is ?? = ?? ?? ?+ ?? ?? ? www.mathssupport.org www.mathssupport.org
Worked example Given the coordinates of two points we can use the formula ?+ ?? ?? ? ?? ?? to directly find the distance between them. For example: What is the distance between the points A(5, 1) and B( 4, 5)? x1 y1 x2 y2 A(5, 1) B( 4, 5) ( 4 5) +(5 2 2 2 2 1) = ( 9) +6 = 81+36 = 117 =3 13 www.mathssupport.org www.mathssupport.org
Worked example Given the coordinates of two points we can use the formula ?+ ?? ?? ? ?? ?? to directly find the distance between them. For example: What is the distance between the points A(6, 3) and B(8, -2)? x1 y1 x2 y2 A(6, 3) B(8, -2) ??+ ?? ? ??+ ? ?? = = ? + ?? = ?? www.mathssupport.org www.mathssupport.org
Distance between two points in space If A = (x1, y1, z1) And G = (x2, y2, z2) z H Find the distance AG Using Pythagoras theorem AG2 = AC2 + CG2 E y z2 G z2 z1 (x2, y2, z2) AC2 = AB2 + BC2 y2 z2 z1 A F D AG2 = AB2 + BC2 + CG2 (x1, y1, z1) C AG= ??2 + ??2 + ??2 B x2 AG= (x2 x1)2+ (y2 y1)2+ (z2 z1)2 x Distance = 2+ ?2 ?1 2+ ?2 ?1 2 ?2 ?1 www.mathssupport.org www.mathssupport.org
Distance between two points in space Find the distance from A = (1, 3, 4) to B = (4, 2, 7) Distance AB = 32+ 12+ 32 = 9 + 1 + 9 = 4.36 (3 sf) = 19 www.mathssupport.org www.mathssupport.org
Distance between two points in space , B=(2, 6, 5) and C=(1, 4, 3) Show that A=(0, 4, 4) Are vertices of an isosceles triangle. Distance AC Distance AB = 2 02+ 6 42+ 5 42 1 02+ 4 42+ 3 42 = 22+ 22+ 12 = 12+ 02+ 12 = = = 4 + 4 + 1 9 = 3 = = 1 + 0 + 1 2 Distance BC Since AB = BC AC ABC is an isosceles triangle. 1 22+ 4 62+ 3 52 = 12+ 22+ 22 = = = 1 + 4 + 4 9 = 3 www.mathssupport.org www.mathssupport.org
A formula for the Mid-Point of AB y Mid-point (x2, y2) B y2 X M X y1 X A (x1, y1) x x1 x2 0 The mid-point is the average of the end points: + + + + x x + + 2 + + 2 y y x x y y 1 2 1 2 is 1 2 1 2 = = = = xM , yM M , or 2 2 www.mathssupport.org www.mathssupport.org
Mid-Point What are the coordinates of the midpoint between the points C(-3, 2) and D(1, 4)? The mid-point is the average of the end points: ??= 3 + 1 2 = 2 2 y 5 D(1, 4) 4 X MP(-1, 3) 3 X C(-3, 2) 2 X = 1 1 ??=2 + 4 1 0 -5 -4 -3 -2 -1 2 4 5 3 x 2 -1 =6 -2 = 3 2 The coordinates of the midpoint are (-1, 3). www.mathssupport.org www.mathssupport.org
Mid-Point What are the coordinates of the midpoint between the points A(-2, -2) and B(1, 3)? y The mid-point is the average of the end points: 4 XB(1, 3) 3 2 ??=1 2 1 MP 1 2,1 2 X 2 = 1 1 0 -5 -4 -3 -2 -1 2 4 5 3 x 2 -1 ??=3 2 2 -2 X A(-2, -2) =1 -3 2 The coordinates of the midpoint are 1 2,1 2 www.mathssupport.org www.mathssupport.org
Mid-Point If A = (x1, y1, z1) and B = (x2, y2, z2) The mid-point for 3D coordinates is ??=?1+ ?2 ??=?1+ ?2 ??=?1+ ?2 2 2 2 ?1+ ?2 2 ,?1+ ?2 2 ,?1+ ?2 2 ?? = www.mathssupport.org www.mathssupport.org
Mid-Point Find the coordinates of the Midpoint between A = (1, 5, 4) and B = (3, 1, 7) ?1+ ?2 2 ,?1+ ?2 2 ,?1+ ?2 2 ?? = 1 + 3 2 ,5 + 1 2 ,4 + 7 2 ?? = MP = (2, 3, 5.5) www.mathssupport.org www.mathssupport.org
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