Question

The point which divides the line joining the points \(\mathrm{A}(1,2)\) and \(\mathrm{B}(-1,1)\) internally in the ratio \(1: 2\) is (1) \(\left(\frac{-1}{3}, \frac{5}{3}\right)\) (2) \(\left(\frac{1}{3}, \frac{5}{3}\right)\) (3) \((-1,5)\) (4) \((1,5)\)

Short answer

Answer: (2) \(\left(\frac{1}{3}, \frac{5}{3}\right)\)

Step-by-step solution

Recall the section formula

To find the point P which divides the line segment AB in the ratio m : n where A(x1, y1) and B(x2, y2), we can use the section formula: P(x, y) = \(\left(\frac{m x_2 + nx_1}{m + n}, \frac{m y_2 + ny_1}{m + n}\right)\) In our case, A(1,2), B(-1,1), and the ratio is 1:2 (m = 1, n = 2).

Apply the section formula to find the required point (P)

Plug in the values of A, B, m and n into the section formula: P(x, y) = \(\left(\frac{1 \times (-1) + 2 \times 1}{1 + 2}, \frac{1 \times 1 + 2 \times 2}{1 + 2}\right)\)

Simplify the point P

Calculate the x and y coordinates of P: P(x, y) = \(\left(\frac{-1 + 2}{3}, \frac{1 + 4}{3}\right)\) P(x, y) = \(\left(\frac{1}{3}, \frac{5}{3}\right)\) Now, we will compare this point with the given options.

Choose the correct option

The point P(x, y) = \(\left(\frac{1}{3}, \frac{5}{3}\right)\), which matches option 2. So, the correct answer is: (2) \(\left(\frac{1}{3}, \frac{5}{3}\right)\)

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is a mathematical discipline that utilizes a coordinate system to define and study the geometric properties of objects. A coordinate system assigns numerical values to each point, allowing us to analyze positions in two or three-dimensional spaces.

• Points: Represented by coordinates (x, y) in a plane.
• Lines: Defined by equations and can be explored through their slopes and intercepts.
• Shapes and Curves: Including circles, ellipses, and more, each with specialized equations.

In coordinate geometry, you can calculate the distance between two points, find the midpoint of a segment, and understand the inclination of a line. It offers powerful tools to visualize and solve complex mathematics problems with precision.

Ratio Division

Ratio division is a concept in mathematics used to divide a line segment into parts that are proportional to given numbers. When a line segment is divided internally by a point, we use the section formula to find this dividing point.
Consider two points A and B dividing a line in a ratio \(m:n\),

• The formula for the x-coordinate of the dividing point offers: \( \frac{mx_2 + nx_1}{m+n} \)
• The formula for the y-coordinate provides: \( \frac{my_2 + ny_1}{m+n} \)

By using these, you can determine the precise location of a point that splits a segment into a desired ratio. It's that simple, yet pivotal, in topics like geometry and physics.

Line Segment

A line segment is a portion of a line confined by two endpoints, having a definite length. Unlike lines, line segments do not extend infinitely in both directions.

• Endpoints: Are fixed at two distinct points on a plane, identified by coordinates.
• Length: Calculated using the distance formula \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).
• Division: Can be divided by other points creating new segments using the section formula.

Line segments are fundamental in geometry for constructing shapes and are used regularly in proofs and theorems. Understanding how to manipulate and measure them is key in solving geometric problems.

Mathematics Problem Solving

Mathematics problem solving harnesses logic, creativity, and knowledge to find solutions. It involves understanding a problem, devising a plan, executing it, and then reflecting on the outcome.

• Understanding: Grasp the problem thoroughly, identifying known and unknown information.
• Planning: Choose strategies, formulas, or theorems that could solve the task.
• Executing: Apply the chosen methods systematically.
• Reflecting: Confirm the solution's validity and consider alternate solutions or errors.

Probleming solving is not only about arriving at the answer but learning the pathway to the solution. It embodies critical thinking, patience, and resilience, forming a core skill in mathematics education.