Introduction to Geometric Transformations and Square Vertex Coordinates

Understanding how to find the coordinates of unknown vertices of geometric shapes like squares based on given information is an essential skill in coordinate geometry. This article will demonstrate the process of finding the coordinates of vertices C and D of square ABCD, given that the coordinates of vertices A and B are 11.5 and 33 respectively. The geometric transformations involved here include reflections and rotations, which help in determining the positions of the missing vertices.

Given Information and Initial Setup

We are given the coordinates of two vertices of a square, A(11.5) and B(33). The task is to find the coordinates of the remaining two vertices, C and D. This problem can be solved using basic coordinate geometry principles and geometric transformations.

First Method: Using Geometric and Parametric Equations

To start with, we can use the vectors and parametric equations of the circle to determine the coordinates of the vertices. We know that the vectors and circles can be used to reflect points about a certain point, which is the origin O in this case. Let's dive into the details:

1. Vector Calculation:
The vector ( overrightarrow{BA} ) can be determined as ( 21.5 0.543 ). This values are derived from the given coordinates of A and B.

2. Parametric Equation of Circle:
The parametric equations of circle centered at A are given by:
( x 2.5 cos alpha_1 )
( y 2.5 sin alpha_1 1.5 )

Similarly, for points D, we can use the angles ( frac{pi}{2} alpha ) and ( frac{3pi}{2} alpha ) to find the coordinates explicitly:

3. Reflection and Rotation:
The coordinates of D can be determined using the angles and trigonometric identities as follows:
( D -1.5121.5 -0.5, 3.5 )
( D 1.51 - 21.5 2.5 - 0.5 )

These steps involve the use of trigonometric functions to rotate the points and reflect them about the origin.

Second Method: Using Coordinate Geometry and Slopes

Another approach is to use coordinate geometry and the properties of slopes of perpendicular lines.
1. Calculate the Distance AB:
[ AB sqrt{(3-1.5)^2 left(frac{3}{2}-frac{3}{2}right)^2} frac{5sqrt{2}}{2} frac{5}{2} ]

2. slope of AB:
[ text{slope of } AB frac{frac{3}{2}-frac{3}{2}}{3-1} frac{3}{4} ]

3. slope of AD:
Since AD is perpendicular to AB, the slope of AD is (-frac{4}{3}).

4. Equation of AD:
Using the point-slope form, the equation of AD is:
[ frac{y-frac{3}{2}}{x-1} -frac{4}{3} ]

5. Solve for Coordinates of D:
By substituting the values and solving, we get two possible coordinates for D: ( left(frac{5}{2}, -frac{1}{2}right) ) and ( left(-frac{1}{2}, frac{7}{2}right) ).

These coordinates help in defining the two possible configurations of square ABCD.

Third Method: Using Position Vectors

Another method involves the use of position vectors and geometric transformations. The position vectors of A and B are given by:

( mathbf{a} 1, frac{3}{2} ), (mathbf{b} 3, 3 )

The vector ( mathbf{p} mathbf{b} - mathbf{a} 2, frac{3}{2} ). To find ( mathbf{q} mathbf{AD} ), where ( mathbf{p} ) and ( mathbf{q} ) are mutually perpendicular and of equal length, we solve the equations:

1. Perpendicularity Condition:
[ 2x frac{3}{2}y 0 ]

2. Length Condition:
[ x^2 left(frac{3}{2}yright)^2 4 left(frac{3}{2}right)^2 frac{25}{4} ]

By solving these, we get ( mathbf{q} ) with possible values for ( t ) as ( frac{1}{2} ) or ( -frac{1}{2} ).

Finally, the coordinates of D and C are determined by vector addition and subtraction, leading to the possible configurations of ABCD and ABCD'.

Conclusion

By using vector algebra, parametric equations, and geometric transformations, we have successfully determined the coordinates of vertices C and D of square ABCD. The problem demonstrates the power of coordinate geometry in solving real-world geometric problems effectively.

Understanding these techniques not only helps in solving specific problems but also in building a strong foundation in geometry and mathematical modeling. For further reading and practice, you might want to explore problems related to geometric transformations and coordinate geometry in various textbooks and online resources.