Plot the points P(1, 0), Q(4,0) and S(1, 3). Find the coordinates of the point R such that PQRS is a square.

Options

A. (2,3)
B. (3,3)
C. (4,3)
D. (0,3)

Answer

To find the coordinates of the point R such that PQRS forms a square, we will follow these steps:

### Step 1: Plot the Points
We start by plotting the given points P(1, 0), Q(4, 0), and S(1, 3) on the Cartesian coordinate system.

- **Point P**: (1, 0) is located on the x-axis at 1 unit from the origin.
- **Point Q**: (4, 0) is also on the x-axis, located at 4 units from the origin.
- **Point S**: (1, 3) is located directly above point P, at 1 unit on the x-axis and 3 units on the y-axis.

### Step 2: Determine the Length of the Square's Side
Next, we need to determine the length of the sides of the square. The distance between points P and S can be calculated as follows:

- The distance between P(1, 0) and S(1, 3) is the vertical distance:
\[ \text{Distance} = |y_2 - y_1| = |3 - 0| = 3 \]

Since PQRS is a square, all sides must be equal. Therefore, the length of each side of the square is 3 units.

### Step 3: Find the Coordinates of Point R
To find point R, we need to ensure that it forms a right angle with points P and Q, and maintains the square's properties.

1. **Coordinates of Q**: (4, 0)
2. **Coordinates of S**: (1, 3)

Since R must be directly above Q to maintain the square's shape, we keep the x-coordinate of Q and use the y-coordinate from S:

- The x-coordinate of R will be the same as Q, which is 4.
- The y-coordinate of R will be the same as S, which is 3.

Thus, the coordinates of point R are:
\[ R(4, 3) \]

### Final Coordinates
The coordinates of the point R such that PQRS forms a square are:
\[ R(4, 3) \]

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