Section and Mid-Point Formula Chapter Notes | Mathematics Class 10 ICSE PDF Download

Introduction

Imagine you’re a treasure hunter on a giant graph paper, where every dot marks a clue to a hidden prize! The "Section and Mid-Point Formula" chapter is your map, guiding you to measure distances, split paths, or find the exact middle of a line. It’s like having a superpower to navigate the world of coordinate geometry with ease. Whether you’re dividing a line into parts or finding a triangle’s center, this chapter makes it simple and fun. Let’s jump into this math adventure and uncover the secrets of points and lines!

Coordinate geometry helps you explore points on a graph (called a Cartesian plane).
It lets you solve problems about lines connecting two points by:

• Finding how far apart two points are.
• Locating a point that splits a line into a specific ratio.
• Finding the middle point of a line.
• Writing the equation of a line passing through two points.
• Figuring out the equation of a line that cuts another line in half at a right angle (perpendicular bisector).

The Section Formula

• This formula finds the coordinates of a point that splits a line segment into a given ratio, like cutting a rope into two pieces of specific lengths.
• If point P divides the line from A(x₁, y₁) to B(x₂, y₂) in the ratio m₁:m₂, it means the length AP:PB = m₁:m₂.
• Formula: Coordinates of P = 

• Steps to find the point:
  • Write the coordinates of points A and B.
  • Check the ratio m₁:m₂ (how the line is divided).
  • Put the values into the formula for x and y coordinates.
  • Simplify to get the coordinates of point P.
• To find the ratio when you know the dividing point:
  • Use the same formula, but set the point’s coordinates equal to x and y.
  • Solve for m₁ and m₂ to find the ratio.
• Alternative Method (k:1 ratio): Assume the ratio is k:1. Then, 
  Solve for k to get the ratio k:1.

Points of Trisection

• Trisection means splitting a line into three equal parts, like cutting a stick into three equal pieces.
• Points P and Q divide line AB so that AP = PQ = QB.
  
• P divides AB in ratio 1:2 (AP:PB).
• Q divides AB in ratio 2:1 (AQ:QB).
• Use the section formula with these ratios to find coordinates of P and Q.
• To check if a point trisects a line, verify if it divides in 1:2 or 2:1.

Example 1: Find the points of trisection of the line joining A(6, -2) and B(-8, 10).

• Step 1: For P (ratio 1:2): x = (1 × -8 + 2 × 6)/(1 + 2) = (-8 + 12)/3 = 4/3.
• Step 2: y = (1 × 10 + 2 × -2)/(1 + 2) = (10 - 4)/3 = 2. P = (4/3, 2).
• Step 3: For Q (ratio 2:1): x = (2 × -8 + 1 × 6)/(2 + 1) = (-16 + 6)/3 = -10/3.
• Step 4: y = (2 × 10 + 1 × -2)/(2 + 1) = (20 - 2)/3 = 6. Q = (-10/3, 6).

Example 2: Show that P(3, m-5) is a point of trisection of the line joining A(4, -2) and B(1, 4). Find m.

• Step 1: P divides AB in 1:2 or 2:1. Try 1:2.
• Step 2: x-coordinate: 3 = (1 × 1 + 2 × 4)/(1 + 2) = (1 + 8)/3 = 3 (correct).
• Step 3: y-coordinate: m - 5 = (1 × 4 + 2 × -2)/(1 + 2) = (4 - 4)/3 = 0.
• Step 4: Solve: m - 5 = 0 → m = 5.
• Step 5: P(3, 0) divides in 1:2, so it’s a trisection point.

Mid-Point Formula

• This formula finds the middle point of a line segment, like finding the center of a bridge.
• For points A(x₁, y₁) and B(x₂, y₂), the midpoint P splits AB in ratio 1:1.
• Formula: Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Steps:
  • Add the x-coordinates of A and B, then divide by 2.
  • Add the y-coordinates of A and B, then divide by 2.
  • The result gives the midpoint’s coordinates.

Example 2: The midpoint of line segment AB is (-3, 5). A lies on the x-axis, B on the y-axis. Find A and B.

• Step 1: Let A = (x, 0), B = (0, y).
• Step 2: Midpoint = ((x + 0)/2, (0 + y)/2) = (-3, 5).
• Step 3: x/2 = -3 → x = -6; y/2 = 5 → y = 10.
• Step 4: A = (-6, 0), B = (0, 10).

Centroid of a Triangle

• The centroid is the point where all three medians (lines from a vertex to the midpoint of the opposite side) of a triangle meet.
• It divides each median in the ratio 2:1 (closer to the vertex).
• Formula: Centroid = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3) for vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃).
• Steps:
  • List the coordinates of vertices A, B, and C.
  • Add all x-coordinates and divide by 3.
  • Add all y-coordinates and divide by 3.
  • The result is the centroid’s coordinates.
• Alternative method:
  • Find the midpoint of one side (e.g., BC).
  • Use the section formula to find the point dividing the median (from A to midpoint of BC) in ratio 2:1.

Example 1: Find the centroid of triangle ABC with vertices A(-2, 3), B(6, 7), C(4, 1).

• Step 1: x = (-2 + 6 + 4)/3 = 8/3.
• Step 2: y = (3 + 7 + 1)/3 = 11/3.
• Step 3: Centroid = (8/3, 11/3).
• Alternative: Midpoint D of BC = ((6 + 4)/2, (7 + 1)/2) = (5, 4).
• Step 4: Centroid G divides AD in 2:1: x = (2 × 5 + 1 × -2)/(2 + 1) = 8/3.
• Step 5: y = (2 × 4 + 1 × 3)/(2 + 1) = 11/3.
• Centroid = (8/3, 11/3).

Example 2: In triangle OBC, O(0, 0), B(-6, 9), C(12, -3). P divides OB in 1:2, Q divides OC in 1:2. Show PQ = 1/3 BC.

• Step 1: For P: x = (1 × -6 + 2 × 0)/(1 + 2) = -6/3 = -2; y = (1 × 9 + 2 × 0)/(1 + 2) = 9/3 = 3. P = (-2, 3).
• Step 2: For Q: x = (1 × 12 + 2 × 0)/(1 + 2) = 12/3 = 4; y = (1 × -3 + 2 × 0)/(1 + 2) = -3/3 = -1. Q = (4, -1).
• Step 3: PQ = √((4 - (-2))² + (-1 - 3)²) = √(6² + (-4)²) = √(36 + 16) = 2√13.
• Step 4: BC = √((12 - (-6))² + (-3 - 9)²) = √(18² + (-12)²) = √(324 + 144) = 6√13.
• Step 5: PQ = 2√13, BC = 6√13 → PQ = 1/3 BC.