Coordinate Geometry - Distance and Section Formula

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Q1. What is the midpoint of the line segment joining the points (4, 5) and (10, 15)?

Solution:

Midpoint = ((4 + 10)/2, (5 + 15)/2) = (7, 10).

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Q2. Find the coordinates of the point that divides the segment joining (2, 3) and (8, 7) 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*8 + 3*2)/(1+3), (1*7 + 3*3)/(1+3)) = (5, 5).

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Q3. If point A(2, 3) and point B(8, 7) are the endpoints of a line segment, what is the midpoint M of AB?

Solution:

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

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Q4. If triangle ABC has vertices A(1, 2), B(4, 6), and C(1, 6), what is the length of side AB?

Solution:

Using the distance formula: AB = √((4 - 1)² + (6 - 2)²) = √(9 + 16) = √25 = 5.0.

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Q5. Find the coordinates of the point that divides the segment joining (1, 2) and (5, 6) in the ratio 2:1.

Solution:

Using the section formula: P = ((2*5 + 1*1)/(2+1), (2*6 + 1*2)/(2+1)) = (3, 4).

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Q6. What is the distance between the points (5, 5) and (5, 1)?

Solution:

Using the distance formula: d = √((5 - 5)² + (1 - 5)²) = √(0 + 16) = √16 = 4.0.

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Q7. What is the area of triangle formed by points A(0, 0), B(4, 0), and C(4, 3)?

Solution:

Area = 0.5 * base * height = 0.5 * 4 * 3 = 6.

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Q8. If point D(3, 4) is the midpoint of segment AB where A(1, 2) and B(x, y), what are the coordinates of B?

Solution:

Using the midpoint formula: D = ((x1 + x2)/2, (y1 + y2)/2). Solving gives B = (5, 6).