Question
The coordinates of three vertices of a square are (-4, -4), (3, -4) and (3, 3). What are the coordinates of the fourth vertex?
Answers
GPT-4o mini
To find the coordinates of the fourth vertex of the square, we can start by identifying the given vertices of the square:
1. Vertex A: (-4, -4)
2. Vertex B: (3, -4)
3. Vertex C: (3, 3)
Since we know three vertices of the square, we can find the fourth vertex (let's call it D) using the properties of squares.
First, let's analyze the vertices we have:
- We can see that points A and B have the same y-coordinate (-4), which means they are horizontal points sharing the same line across the x-axis. The distance between them can be calculated:
\[
\text{Distance AB} = x_B - x_A = 3 - (-4) = 3 + 4 = 7
\]
- Points B and C have the same x-coordinate (3), which means they are vertical points sharing the same line along the y-axis. The distance between them can be calculated:
\[
\text{Distance BC} = y_C - y_B = 3 - (-4) = 3 + 4 = 7
\]
Next, since we have two distances of 7, we know that the sides of the square are equal and both distances mentioned are correct as they confirm the square's properties.
Now, we can find point D which we will derive by applying the rules of a square. The coordinates of the fourth vertex can be either depending on symmetry.
Since we have:
- To get from A to C \( \Delta y \) changes from -4 to 3 (increases), while changing \(-4\) to \(3\) means \( \Delta x \) must support an extension in the negative x-axis for the fourth vertex analysis.
The coordinates of D can be evaluated based on the observations we have. The x-coordinate of D would be the same as A (-4) and the y-coordinate will be the same as C (3).
So the coordinates of D are:
\[
D: (-4, 3)
\]
Thus, the coordinates of the fourth vertex are \(\boxed{(-4, 3)}\).
1. Vertex A: (-4, -4)
2. Vertex B: (3, -4)
3. Vertex C: (3, 3)
Since we know three vertices of the square, we can find the fourth vertex (let's call it D) using the properties of squares.
First, let's analyze the vertices we have:
- We can see that points A and B have the same y-coordinate (-4), which means they are horizontal points sharing the same line across the x-axis. The distance between them can be calculated:
\[
\text{Distance AB} = x_B - x_A = 3 - (-4) = 3 + 4 = 7
\]
- Points B and C have the same x-coordinate (3), which means they are vertical points sharing the same line along the y-axis. The distance between them can be calculated:
\[
\text{Distance BC} = y_C - y_B = 3 - (-4) = 3 + 4 = 7
\]
Next, since we have two distances of 7, we know that the sides of the square are equal and both distances mentioned are correct as they confirm the square's properties.
Now, we can find point D which we will derive by applying the rules of a square. The coordinates of the fourth vertex can be either depending on symmetry.
Since we have:
- To get from A to C \( \Delta y \) changes from -4 to 3 (increases), while changing \(-4\) to \(3\) means \( \Delta x \) must support an extension in the negative x-axis for the fourth vertex analysis.
The coordinates of D can be evaluated based on the observations we have. The x-coordinate of D would be the same as A (-4) and the y-coordinate will be the same as C (3).
So the coordinates of D are:
\[
D: (-4, 3)
\]
Thus, the coordinates of the fourth vertex are \(\boxed{(-4, 3)}\).