Coordinate Geometry - Distance and Section Formula

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Q1. If point A(2, 3) and point B(8, 7) are given, what is the midpoint M of segment AB?

Solution:

Using the midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

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Q2. Find the distance between the points (-1, -1) and (2, 3).

Solution:

Using the distance formula: d = √((2 - (-1))² + (3 - (-1))²) = √((3)² + (4)²) = √(9 + 16) = √25 = 5.

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Q3. Find the coordinates of the point that divides the line segment joining (3, 4) and (9, 10) in the ratio 1:1.

Solution:

Using the section formula: P = ((1*9 + 1*3)/(1+1), (1*10 + 1*4)/(1+1)) = (6, 7).

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Q4. What is the coordinates of the point that divides the line segment joining (0, 0) and (4, 4) in the ratio 1:3?

Solution:

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) where m=1, n=3, P = ((1*4 + 3*0)/(1+3), (1*4 + 3*0)/(1+3)) = (1, 1).

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Q5. What is the section formula for dividing the line segment joining points (1, 2) and (4, 6) in the ratio 2:1?

Solution:

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) where m=2, n=1, P = ((2*4 + 1*1)/(2+1), (2*6 + 1*2)/(2+1)) = (3, 4).

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Q6. If point C(1, 2) and point D(7, 8) are given, what is the coordinates of the point that divides CD in the ratio 3:2?

Solution:

Using the section formula: P = ((3*7 + 2*1)/(3+2), (3*8 + 2*2)/(3+2)) = (4, 5).