Master Coordinate Geometry in 30 Minutes: A Quick Guide for CBSE Class 10 Students
Q&A Style Revision
1. What is the Distance Formula in Coordinate Geometry?
The distance between two points ((x_1, y_1)) and ((x_2, y_2)) is given by:
(d = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2})
2. How do you calculate the Midpoint of a line segment?
The midpoint of a line segment joining two points ((x_1, y_1)) and ((x_2, y_2)) is:
(left(frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2}right))
3. What is the formula for the Section Formula?
If a line segment joining points ((x_1, y_1)) and ((x_2, y_2)) is divided by a point in the ratio (m:n), the coordinates of the point are:
(left(frac{mx_2 + nx_1}{m+n}, frac{my_2 + ny_1}{m+n}right))
4. How do you find the Area of a Triangle using Coordinate Geometry?
The area of a triangle with vertices ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)) is:
(text{Area} = frac{1}{2}left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)right|)
5. Solve: Find the distance between the points ((3, 4)) and ((7, 1)).
Solution:
- Use the distance formula: (d = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}).
- (d = sqrt{(7 – 3)^2 + (1 – 4)^2} = sqrt{4^2 + (-3)^2} = sqrt{16 + 9} = sqrt{25} = 5).
Distance: 5 units
6. Solve: Find the midpoint of a line segment joining ((2, -3)) and ((-4, 5)).
Solution:
- Use the midpoint formula: (left(frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2}right)).
- (text{Midpoint} = left(frac{2 + (-4)}{2}, frac{-3 + 5}{2}right) = left(frac{-2}{2}, frac{2}{2}right) = (-1, 1)).
Midpoint: (-1, 1)
7. Solve: Find the coordinates of the point dividing the line segment joining ((1, 2)) and ((3, 6)) in the ratio 1:2.
Solution:
- Use the section formula: (left(frac{mx_2 + nx_1}{m+n}, frac{my_2 + ny_1}{m+n}right)).
- (text{Point} = left(frac{1cdot 3 + 2cdot 1}{1+2}, frac{1cdot 6 + 2cdot 2}{1+2}right) = left(frac{3+2}{3}, frac{6+4}{3}right) = left(frac{5}{3}, frac{10}{3}right)).
Coordinates: (left(frac{5}{3}, frac{10}{3}right))
8. Solve: Find the area of a triangle with vertices ((0, 0)), ((4, 0)), and ((0, 3)).
Solution:
- Use the area formula: (text{Area} = frac{1}{2}left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)right|).
- (text{Area} = frac{1}{2}left|0(0-3) + 4(3-0) + 0(0-0)right| = frac{1}{2}left|0 + 12 + 0right| = frac{1}{2}cdot 12 = 6).
Area: 6 square units