Section Formula for Internal Division
To find the point that divides a line segment internally in a given ratio, we use the section formula.
Let the two points be \(P(x_1, y_1) = (10, 18)\) and \(Q(x_2, y_2) = (5, 8)\). Let the ratio be \(m : n = 2 : 3\).
The coordinates \((x, y)\) of the point dividing the line segment internally are given by the section formula:
\(x = \frac{mx_2 + nx_1}{m+n}\) \(y = \frac{my_2 + ny_1}{m+n}\)Calculating Coordinates
Substitute the given values into the formula:
X-coordinate calculation:
\(x = \frac{(2 \times 5) + (3 \times 10)}{2+3}\) \(x = \frac{10 + 30}{5}\) \(x = \frac{40}{5}\) \(x = 8\)Y-coordinate calculation:
\(y = \frac{(2 \times 8) + (3 \times 18)}{2+3}\) \(y = \frac{16 + 54}{5}\) \(y = \frac{70}{5}\) \(y = 14\)Therefore, the point dividing the line segment internally is \((8, 14)\).
Comparing with Options
The calculated point \((8, 14)\) matches Option 3.
Similar Questions
- \(A(1, 2)\), \(B(a, -26)\), \(C(13, -14)\) and \(D(6, 14)\) are the vertices of a parallelogram, taken in order, find the value of \(a\).
- What would be the area of the triangle with \(A(8, 8)\), \(B(22, 8)\) and \(C(16, -10)\) as vertices?
- Find a point on the X axis which is equidistant from A(2, -3) and B(-2, 1)
- What is the midpoint of the line segment joining \((-2, -1)\) and \((-5, -3)\)?
- If point C is equidistant from A(5, -6) and B(7, 8), coordinate of C will be
- The area of the triangle whose vertices are given by (2, 4), (– 3, – 1) and (5, 3) is:
Important Questions from Coordinate Geometry
Find the co-ordinates of the centroid of a triangle whose vertices are A(1, 4), B(7, 8) and C(10, 12).
- If x² + y² - 16x + 38y + 425 = 0, then the value of x² + y² is:
- If x² + y² - 12x + 18y + 117 = 0, then the value of x² + y² is:
- A geological survey team has established three research stations forming a triangular monitoring zone. The stations are located at coordinates P(2,5), Q(8,1), and R(4,9) on a coordinate grid where each unit represents 1 kilometer. What is the area of the triangular monitoring zone?
- \(A(1, 2)\), \(B(a, -26)\), \(C(13, -14)\) and \(D(6, 14)\) are the vertices of a parallelogram, taken in order, find the value of \(a\).
