Coordinate Geometry - Distance and Section Formula - Coordinate Geometry Applications

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Q1. If point A is at (3, 5) and point B is at (9, 1), what is the coordinates of point P that divides AB in the ratio 2:3?

Solution:

Using the section formula: P = ((2*9 + 3*3)/(2+3), (2*1 + 3*5)/(2+3)) = (5.4, 3.6).

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Q2. What is the midpoint of the line segment joining the points (2, 3) and (8, 7)?

Solution:

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

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Q3. If point C is at (3, 5) and point D is at (9, 1), what is the coordinates of the point that divides CD in the ratio 2:3?

Solution:

Using the section formula: P = ((2*9 + 3*3)/(2+3), (2*1 + 3*5)/(2+3)) = (5.4, 3.2).

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Q4. What is the coordinates of the point that divides the line segment joining (2, 3) and (4, 7) in the ratio 3:1?

Solution:

Using the section formula: P = ((3*4 + 1*2)/(3+1), (3*7 + 1*3)/(3+1)) = (3.5, 5).

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Q5. If point A is at (1, 2) and point B is at (4, 6), what is the section formula for point P that divides AB in the ratio 1:2?

Solution:

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

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Q6. What is the coordinates of the point that divides the line segment joining (6, 8) and (2, 4) in the ratio 1:1?

Solution:

Using the section formula: P = ((1*2 + 1*6)/(1+1), (1*4 + 1*8)/(1+1)) = (4, 6).

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Q7. What is the distance between the points (1, 1) and (1, 5)?

Solution:

Using the distance formula: d = √((1 - 1)² + (5 - 1)²) = √(0 + 16) = √16 = 4.

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Q8. What is the distance between the points (0, 0) and (5, 12)?

Solution:

Using the distance formula: d = √((5 - 0)² + (12 - 0)²) = √(25 + 144) = √169 = 13.0.