Section Formula (Internal Division) MCQs Quiz | Class 10
This quiz focuses on Class X Mathematics, specifically Unit III: Coordinate Geometry. Test your understanding of the Section Formula for Internal Division, including finding the coordinates of a point that divides a line segment internally in a given ratio and various numerical applications. Complete the quiz and submit your answers, then download your personalized answer sheet in PDF format.
Understanding the Section Formula (Internal Division)
The Section Formula is a fundamental concept in coordinate geometry that helps us find the coordinates of a point that divides a line segment in a given ratio. This formula is particularly useful in various geometric problems and is an essential tool for Class 10 students.
What is Internal Division?
When a point lies between two other points on a line segment and divides it into two smaller segments, it is said to divide the line segment internally. The ratio of these two smaller segments is crucial for applying the section formula.
The Section Formula
Suppose we have a line segment joining two points A(x1, y1) and B(x2, y2). If a point P(x, y) divides this line segment internally in the ratio m1:m2 (meaning AP:PB = m1:m2), then the coordinates of P are given by:
P(x, y) = (`(m1*x2 + m2*x1) / (m1 + m2)`, `(m1*y2 + m2*y1) / (m1 + m2)`)
Key Points to Remember:
- Understanding the Ratio: The ratio m1:m2 indicates how the segment is divided. m1 is associated with the segment from A to P, and m2 with the segment from P to B.
- Order of Points Matters: Ensure you correctly identify (x1, y1) as the coordinates of the first point and (x2, y2) as the coordinates of the second point.
- Special Case – Midpoint Formula: When a point divides a line segment in the ratio 1:1, it is the midpoint. In this case, m1 = m2 = 1. The midpoint formula is a simplified version of the section formula:
Midpoint M(x, y) = (`(x1 + x2) / 2`, `(y1 + y2) / 2`)
- Applications: The section formula is used to find the centroid of a triangle (which divides the median in a 2:1 ratio), to check collinearity, and to solve various problems involving division of line segments.
How to Apply the Section Formula (Step-by-Step)
| Step | Description | Example (A(1,2), B(7,8), ratio 1:2) |
|---|---|---|
| 1. Identify Points | Assign (x1, y1) and (x2, y2) to the endpoints. | (x1, y1) = (1, 2), (x2, y2) = (7, 8) |
| 2. Identify Ratio | Determine the internal division ratio m1:m2. | m1 = 1, m2 = 2 |
| 3. Calculate x | Use x = (m1x2 + m2x1) / (m1 + m2) | x = (1*7 + 2*1) / (1 + 2) = (7 + 2) / 3 = 9/3 = 3 |
| 4. Calculate y | Use y = (m1y2 + m2y1) / (m1 + m2) | y = (1*8 + 2*2) / (1 + 2) = (8 + 4) / 3 = 12/3 = 4 |
| 5. State Coordinates | The coordinates of the dividing point P(x, y). | P(3, 4) |
Quick Revision Checklist:
- Formula: Remember P(x, y) = ((m1x2 + m2x1) / (m1 + m2), (m1y2 + m2y1) / (m1 + m2))
- Midpoint: A special case where m1 = m2 = 1.
- Ratio k:1: Sometimes, problems are easier by assuming a ratio k:1. If k is positive, it’s internal division.
- Axes Division:
- Point on x-axis: (x, 0) (use y-coordinate for ratio calculation)
- Point on y-axis: (0, y) (use x-coordinate for ratio calculation)
Practice Questions:
- Find the coordinates of the point which divides the line segment joining (-3, 5) and (4, -9) in the ratio 3:4 internally.
- In what ratio does the point (-4, 6) divide the line segment joining A(-6, 10) and B(3, -8)?
- If P(1, -2) divides the line segment joining A(3, -6) and B(x, y) in the ratio 1:2 internally, find the coordinates of B.
- Find the coordinates of the vertices of a triangle if the midpoints of its sides are (1, -1), (-4, 2), and (-3, -5).
- The vertices of a triangle are A(2, 1), B(5, 2), and C(3, 4). Find the coordinates of the centroid of the triangle.
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