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Chapter Analysis
Intermediate16 pages • EnglishQuick Summary
This chapter introduces the concepts of coordinate geometry, focusing on distance formula, section formula, and applications of coordinate geometry. It elaborates on how to find the distance between two points given their coordinates and how to determine the coordinates of a point dividing a line segment in a given ratio. Various examples and exercises solidify understanding, making these concepts accessible and practical.
Key Topics
- •Distance formula
- •Section formula
- •Midpoint formula
- •Collinearity of points
- •Area of a triangle using coordinates
- •Properties of parallelograms in coordinate geometry
Learning Objectives
- ✓Understand and apply the distance formula to find the distance between two points.
- ✓Utilize the section formula to find coordinates dividing a line segment in a given ratio.
- ✓Determine the midpoint of a line segment using its endpoints.
- ✓Apply knowledge of coordinate geometry to verify collinearity and properties of triangles.
- ✓Solve problems related to the division of line segments both internally and externally.
- ✓Use coordinate geometry to prove properties of geometric shapes like parallelograms.
Questions in Chapter
Find the distance between the following pairs of points: (2, 3), (4, 1) and (– 5, 7), (– 1, 3).
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If Q(0, 1) is equidistant from P(5, –3) and R(x, 6), find the values of x. Also find the distances QR and PR.
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Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (– 3, 4).
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Additional Practice Questions
Determine the mid-point of the line segment joining the points (6, -3) and (-4, 7).
easyAnswer: The mid-point is ((6 + (-4))/2, (-3 + 7)/2) = (1, 2).
A triangle has vertices at (1, 2), (3, 4), and (5, 6). Calculate the perimeter of the triangle.
mediumAnswer: Using the distance formula for each side: AB = √((3-1)^2 + (4-2)^2) = √8, BC = √((5-3)^2 + (6-4)^2) = √8, CA = √((5-1)^2 + (6-2)^2) = √32. The perimeter = √8 + √8 + √32 ≈ 4√2 + 4√2 + 5√2 = 13√2.
What are the coordinates of a point that divides the line segment joining (2, 1) and (-2, -3) externally in the ratio 2:3?
hardAnswer: Using the section formula for external division, the coordinates are ((2*-2 + 3*2)/(2-3), (2*-3 + 3*1)/(2-3)) = ((-4 + 6)/-1, (-6 + 3)/-1) = (-2, 3).
If a quadrilateral has vertices at (1, 2), (3, 5), (6, 1), and (4, -2), is it a parallelogram?
mediumAnswer: Calculate the midpoints of diagonals to verify if they bisect each other: M1 for AC = ((1+6)/2, (2+1)/2) = (3.5, 1.5), M2 for BD = ((3+4)/2, (5-2)/2) = (3.5, 1.5). Since the midpoints are equal, it is a parallelogram.
Verify if the points (3, 3), (6, 6), (9, -3) are collinear.
easyAnswer: The points are collinear if the area of the triangle they form is zero. Using the determinant method, the area is 0.5*[3(6+3) + 6(-3-3) + 9(3-6)] = 0. Hence, they are collinear.