Geometry and Coordinate Problems: Solutions and Analysis

In the realm of coordinate geometry, problems related to the coordinates of points and their relationships form the backbone of many complex problem-solving scenarios. In this article, we will explore and solve a series of geometry and coordinate problems, providing a detailed analysis and step-by-step solutions.

Problem 1: Finding the Coordinates of Point C in Line Segment AB

Problem: The coordinates of points A and B are respectively 4, -3 and -17. We need to find the coordinates of a point C in the line segment AB such that AC:CB 3:2.

Solution:

First, let's determine the coordinates of vector (overrightarrow{B-A}): (-17-4 -21, -3-(-3) 0). This gives us (overrightarrow{B-A} (-21, 0)). Next, we use the section formula to find the coordinates of point C. The formula for point C dividing the line segment AB in the ratio 3:2 is given by: (C A frac{3}{5}overrightarrow{B-A}). Substituting the values:

(C (4, -3) frac{3}{5}(-21, 0) (4 - frac{63}{5}, -3) (frac{20 - 63}{5}, -3) (frac{-43}{5}, -3) (1, -3)).

Thus, the coordinates of point C are (1, -3).

Problem 2: Solving a Line Equation and Finding Point C

Problem: Line AB is given by the equation y - 7 -2x - 1, or 2x y 5. We need to find the coordinates of point C such that 2AC 3BC.

Solution:

First, let's express the coordinates of C in terms of a. We know that C divides the line segment AB in the ratio 2:3. We can use the section formula to find the coordinates of C in terms of a:

(2(4 - a)^2 3(2a - 8)^2 5a^2 10a 5)

Simplifying, we get:

(8(4 - a)^2 9(2a - 8)^2 45a^2 90a 45)

(8(16 - 8a a^2) 9(4a^2 - 32a 64) 45a^2 90a 45)

(128 - 64a 8a^2 36a^2 - 288a 576 45a^2 90a 45)

(44a^2 - 352a 704 45a^2 90a 45)

(-a^2 - 442a 659 0)

Solving the quadratic equation:

(a frac{442 pm sqrt{442^2 - 4(-1)(659)}}{-2} frac{442 pm sqrt{195364 2636}}{-2} frac{442 pm sqrt{198000}}{-2})

(a -11, a 1)

The coordinates of C are (-11, 27) or (1, 3). The coordinate (1, 3) is valid, so the answer is (1, 3).

Problem 3: Using the Section Formula for Point C

Problem: Let the coordinates of point C be (x, y). Using the section formula, find the coordinates of C such that the ratio of segment AC to BC is 3:2.

Solution:

The section formula for point C is given by the formula: (x frac{3(1) - 2(4)}{3 2} frac{3 - 8}{5} -1), (y frac{3(7) - 2(3)}{3 2} frac{21 - 6}{5} 3).

Thus, the coordinates of point C are (-1, 3).

By applying the section formula and solving the equations step-by-step, we can find the coordinates of points in line segments and verify the relationships between them.