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Magazine Chapter 10: Problem 12
Points \(A\) and \(B\) have symmetry with respect to point \(C(2,3)\) Find the coordinates of \(B\) if \(A\) is the point: a) \(\quad(3,-4)\) b) \(\quad(0,2)\) c) \((5,0)\) d) \(\quad(a, b)\)
Short Answer
Expert verified
For a) B(1,10), b) B(4,4), c) B(-1,6), d) B(4-a,6-b).
Step by step solution
01
Understanding symmetry with respect to a point
When two points, A and B, are symmetrical with respect to another point C, this means C is the midpoint of the line segment AB. Therefore, if A is \((x_1, y_1)\) and B is \((x_2, y_2)\), then C, being the midpoint, satisfies the conditions: \(2 = \frac{x_1 + x_2}{2}\) and \(3 = \frac{y_1 + y_2}{2}\).
02
Solve part a: Finding coordinates of B given A(3,-4) and C(2,3)
With A(3, -4) and C(2, 3), plug into the midpoint formula: \(2 = \frac{3 + x_2}{2}\) which solves to \(x_2 = 1\). Also, \(3 = \frac{-4 + y_2}{2}\) which solves to \(y_2 = 10\). Thus, the coordinates of B are (1, 10).
03
Solve part b: Finding coordinates of B given A(0,2) and C(2,3)
With A(0, 2) and C(2, 3), use the midpoint formula: \(2 = \frac{0 + x_2}{2}\) which solves to \(x_2 = 4\). For the y-coordinate: \(3 = \frac{2 + y_2}{2}\), solving gives \(y_2 = 4\). Thus, B is (4, 4).
04
Solve part c: Finding coordinates of B given A(5, 0) and C(2,3)
Given A(5, 0) and C(2, 3), substitute into the midpoint formula: \(2 = \frac{5 + x_2}{2}\) which solves to \(x_2 = -1\). For the y-coordinate: \(3 = \frac{0 + y_2}{2}\), solving gives \(y_2 = 6\). Therefore, B is (-1, 6).
05
Solve part d: Finding coordinates of B given A(a, b) and C(2,3)
Using the general form of midpoint with A(a, b), and C(2, 3), we have: \(2 = \frac{a + x_2}{2}\), which solves to \(x_2 = 4 - a\). Also, \(3 = \frac{b + y_2}{2}\) which solves to \(y_2 = 6 - b\). Thus, B is \((4-a, 6-b)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Midpoint Formula
The midpoint formula is a fundamental tool in coordinate geometry that allows us to calculate the midpoint of a line segment when the coordinates of the endpoints are known. This formula is especially useful when dealing with symmetrical points on a coordinate plane.
The formula to find the midpoint, \( C(x, y) \), of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
This equation essentially averages the x-coordinates and y-coordinates of the endpoints, providing the exact center point of the segment.
The formula to find the midpoint, \( C(x, y) \), of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
This equation essentially averages the x-coordinates and y-coordinates of the endpoints, providing the exact center point of the segment.
- For example, if point A is at (3, -4) and the midpoint C is at (2, 3), applying the formula helps us find point B.
- The simplicity of this formula makes it a go-to method for solving many geometry problems involving symmetrical points.
- When understanding symmetry in geometry, the midpoint often serves as a pivotal point of reference, helping ensure consistency and accuracy in calculations.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves describing geometrical shapes and figures using a coordinate system. This approach allows for a precise method of defining and calculating distances, midpoints, and the relationships between different points.
Some essential features of coordinate geometry include:
Some essential features of coordinate geometry include:
- Point Representation: Each point in a plane is represented with an ordered pair (x, y), providing a clear and measurable means to work with shapes.
- Distance Formula: This calculates the distance between two points, which is crucial in establishing geometric properties.
- Equations of Lines: Coordinate geometry allows us to write the equation of a line in various forms - slope-intercept, point-slope, and more.
Symmetrical Points
Symmetrical points around a reference point (or midpoint) maintain equal distances from that point, forming a balanced configuration. Understanding symmetry involves recognizing how each point is a mirror image of another with respect to a central point.
The concept of symmetry in points is not just aesthetically pleasing but also helps solve mathematical problems efficiently.
The concept of symmetry in points is not just aesthetically pleasing but also helps solve mathematical problems efficiently.
- If point A and point B are symmetrical with respect to midpoint C, this indicates that C is equidistant from A and B. This is why C can be the midpoint of line segment AB, as shown by the midpoint formula.
- The symmetry concept aids in finding coordinates of unknown points when given one symmetrical pair and the midpoint, as demonstrated in problems where you're asked to find point B given point A and midpoint C.
- In the provided exercise, by using symmetry, one can reverse-engineer unknown coordinates by ensuring the same coordinates distance from the midpoint, thus maintaining geometrical balance.
