The Ambiguity of Two Given Points for Square Construction

When given the coordinates of two points on a square, there can be three different squares that can be constructed based on the configuration of these points. This analysis dives into the complexities and solutions for drawing three different squares from two given corners.

Introduction to Square Construction

Solving geometric problems often involves understanding the relationship between different points on a shape. When given the coordinates of two points, such as -44 and 1-1, multiple squares can be constructed. This article explores the three different scenarios and provides detailed solutions for each case.

Case 1: Diagonally Opposite Corners

In the first scenario, the two given points are the endpoints of one diagonal of the square. These points are -44 and 1-1. To solve for the coordinates of the other two vertices (A and C), we follow the steps below:

Step 1: Find the Midpoint of the Diagonal

The midpoint O of the diagonal BD is calculated as (-1.5, 1.5). Since the diagonals of a square bisect each other, the midpoint of AC is also (-1.5, 1.5). The slope of BD is -1, so the slope of AC is 1. The equation of AC, passing through (-1.5, 1.5), is x - y 3.

Step 2: Solve for the Coordinates of A and C

If A is (p, q), then q p - 3 from the equation of AC. The length of the diagonal BD is √50, so the length of AO is 1/2√50. Using the distance formula, p - 1.52 q - 1.52 12.5 and substituting q p - 3. This results in the equation: p2 - 3p 4.52 12.5. Solving the quadratic equation p2 2.52 gives p -2.5 or 1.

Conclusion

Therefore, A is (1, 4) and C is (-4, -1). QED.

Case 2: Consecutive Vertices

In the second scenario, the two given points are consecutive vertices of the square. The steps are as follows:

Step 1: Find the Equation of the Line AB

The equation of line AB (given points A and B) is xy 0. Consecutive vertices means CD is parallel to AB, so its equation is also of the form xy k.

Step 2: Calculate the Distance Between Lines AB and CD

The distance between AB and CD is equal to the length of AB, which is √50. The perpendicular distance from (-1, -1) to line CD is 1-1-k / √2 -5√2. Solving for k gives k -10, so CD is x y -10.

Step 3: Find the Equations of BC and AD

Equation of BC is x - y p, and passing through B(-4, -1), BC is x - y -8. Equation of AD is x - y t, and passing through A(1, -1), AD is x - y 2.

Step 4: Solve for the Coordinates of C and D

Solving for C, we get (9, 1) or (-1, -9). Solving for D, we get (6, 4) or (-4, -6).

Conclusion

QED.

Statement on Ambiguous Corners

The coordinates of two corners of a square being given is an ambiguous problem. There are three possibilities to consider:

Possibility 1: Adjacent Corners of an Edge

The two given points are the endpoints of one side of the square, and two more squares can be constructed with other vertices in the 1st and 3rd quadrants.

Possibility 2: Diagonally Opposite Corners on a Face

The two given points are diagonally opposite corners of the square, and the third square can be constructed with the other two vertices in the 1st and 3rd quadrants.

Possibility 3: Diagonally Opposite Corners of Opposite Sides

The two given points are not directly connected, forming horizontal or vertical sides, and the vertices lie in different quadrants.

Hence, three possible squares can be constructed from two given corners, depending on the configuration of these points.

Conclusion

Given two points, the construction of a square can be ambiguous, leading to three different solutions. Understanding the different configurations of these points is crucial for constructing multiple squares. This detailed analysis provides a clear method for solving such geometric problems.