Distance Between Two Points and Coordinates of Points of Internal and External Division (Distance Formula and Section Formula)
👉 Distance Between Two Points (Distance Formula)
🔹 Distance Between Two Points
⭐ For any two points P(x1, y1) and Q(x2, y2), the distance between them is
🔹 Distance of a Point from the Origin
⭐ If the origin is P(0, 0) and another point is Q(x1, y1), then the distance between them is
Solve the following problems using the Distance Formula.
🔹 Find the distance between the points (3, -5) and (5, -1).
🔹 If the distance between the points (a, -2) and (0, 0) is √29 units, find the possible value or values of a.
🔹 Prove that the points (7, 10), (-2, 5) and (3, -4) are the vertices of an isosceles right-angled triangle.
🔹 Two vertices of an isosceles triangle are (2, 0) and (2, 5). If the equal sides are each 3 units long, find the third vertex.
🔹 Prove that the distance between the given two points is independent of the value of a.
👉 Coordinates of a Point Dividing a Line Segment in a Given Ratio
Coordinates of a Point Dividing a Line Segment Internally
⭐ Suppose a point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio m : n.
∴ and
Coordinates of a Point Dividing a Line Segment Externally
⭐ Suppose a point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2) externally in the ratio m : n.
∴ and
Coordinates of the Midpoint
∴ and
🔹 The midpoint is the special case of internal division where P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio m : n = 1 : 1.
Solve the following problems using the Section Formula.
🔹 Find the coordinates of the point that divides the line segment joining (8, 9) and (–7, 4) internally in the ratio 2 : 3.
🔹 In what ratio does the point P(–1, –1) divide the line segment joining A(4, 4) and B(7, 7)?
🔹 In what ratio does the x-axis divide the line segment joining (–2, 5) and (1, –9)?
🔹 Find the point or points of trisection of the line segment joining (1, 2) and (11, 9).
👉 Important Coordinate Results for a Triangle
Coordinates of the Centroid of a Triangle
⭐ If the vertices of a triangle are A(x1, y1), B(x2, y2) and C(x3, y3), then the centroid is G(x, y) where
∴ and
📚 The point where the three medians of a triangle intersect is called the centroid of the triangle.
📚 The centroid divides each median in the ratio AG : GL = 2 : 1, measured from the vertex.
Coordinates of the Incenter of a Triangle
⭐ Let the vertices of a triangle be A(x1, y1), B(x2, y2) and C(x3, y3). If the sides opposite these vertices are a, b, and c respectively, then the incenter I(x, y) is given by
∴ and
📚 The point where the three internal angle bisectors of a triangle meet is called the incenter.
👉 Area of a Triangle from the Coordinates of Its Vertices
Area of a Triangle
⭐ If the vertices of a triangle are A(x1, y1), B(x2, y2) and C(x3, y3), then its area is
∴
Condition for Three Points to Be Collinear
🔹 If three points lie on the same straight line, the area of the triangle formed by them is zero.
∴
🧠 Quick Formula Revision
| Topic | Formula / Result |
|---|---|
| Distance between two points | |
| Distance from the origin | |
Internal division, m:n | |
External division, m:n | |
| Midpoint | |
| Centroid |
📚 Important Points
🔹 In the Distance Formula, it is convenient to subtract the x-coordinates and y-coordinates in the same order. Since the differences are squared, reversing the order does not change the final distance.
🔹 In the Internal Section Formula, the denominator is m+n, whereas in the External Section Formula, the denominator is m-n. Remember this difference carefully.
🔹 The midpoint formula is a special case of the Section Formula for m:n = 1:1.
🔹 The centroid divides every median in the ratio 2:1 from the vertex.
🔹 The area formula uses an absolute value, so the area cannot be negative.
🔹 If three points are collinear, the area of the triangle formed by them is zero.
✏️ Solved Examples
Example 1: Distance Between Two Points
Find the distance between A(2, 3) and B(8, 11).
Therefore, AB = 10 units.
Example 2: Internal Division
A point P divides the line segment joining A(2, 4) and B(8, 10) internally in the ratio 1:2. Find the coordinates of P.
Therefore, P(4, 6).
Example 3: Testing Collinearity
Check whether the points A(1, 2), B(3, 6) and C(5, 10) are collinear.
Therefore, the three points are collinear.
