To find out the coordinates of the fourth corner D of a square ABCD, we need to understand a few properties of a square and apply them to the given points.
Properties of a square relevant to this problem are:
1. All four sides of a square are equal in length.
2. Adjacent sides are perpendicular to each other.
3. The diagonals of a square bisect each other.
Given the points A(1,5), B(5,5), and C(5,1), we can infer the following:
Since the points A (1, 5) and B (5, 5) both have a y-coordinate of 5, AB is a horizontal line segment. Similarly, the points B (5, 5) and C (5, 1) both have an x-coordinate of 5, BC is a vertical line segment. Therefore, AB is perpendicular to BC. Since the sides are perpendicular and we are dealing with a square, we know that AB and BC are two adjacent sides of the square.
To find the fourth corner D, we can now follow these steps:
Step 1: Calculate the lengths of AB and BC.
- Length of AB = Bx - Ax = 5 - 1 = 4
- Length of BC = By - Cy = 5 - 1 = 4
Here, the x-coordinate and y-coordinate are labeled according to their points, for example, Bx and By are the x and y coordinates of point B.
Since both lengths are equal, we can confirm the shape as a square (given the other properties hold true).
Step 2: The square's fourth corner D will be directly opposite corner A, which means AD and BC (AC and BD) should be parallel, and AD should be equal to BC.
Since AB and BC are sides of a square and AB is a horizontal line segment, AD will also be a horizontal segment equal in length to AB and starting from point A. Since AD is horizontal, the x-coordinate of D will be Ax + length of AD, which is 1 + 4 = 5.
Step 3: Similarly, CD will be a vertical segment parallel and equal in length to BC (and AB). So the y-coordinate of D will be Ay - length of CD, which is 5 - 4 = 1.
Thus, the coordinates of point D, the fourth corner of the square, will be D(5,1).
