The co-ordinates of vertices P and Q of an equilateral `Delta PQR` are `(1, sqrt(3))` and (0, 0). Which of the following could be co-ordinates of R ?

Options

A. `(1, 2)`
B. `(2, 0)`
C. `(1, (sqrt(3))/(2))`
D. `(sqrt(3), 1)`

Answer

To find the coordinates of vertex R of the equilateral triangle PQR, given the coordinates of vertices P and Q, we can follow these steps:

### Step 1: Identify the given coordinates
The coordinates of vertices P and Q are:
- P = (1, √3)
- Q = (0, 0)

### Step 2: Calculate the length of side PQ
Using the distance formula, we can calculate the length of side PQ:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of P and Q:
\[ PQ = \sqrt{(0 - 1)^2 + (0 - \sqrt{3})^2} = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 3: Set the length of sides equal
Since triangle PQR is equilateral, all sides are equal. Therefore, the lengths of sides QR and RP must also equal 2.

### Step 4: Set up the equations for QR and RP
Let the coordinates of R be (x, y). We can set up the equations for the lengths QR and RP.

1. For QR:
\[ QR = \sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2} \]
Setting this equal to 2:
\[ \sqrt{x^2 + y^2} = 2 \implies x^2 + y^2 = 4 \quad \text{(Equation 1)} \]

2. For RP:
\[ RP = \sqrt{(x - 1)^2 + (y - \sqrt{3})^2} \]
Setting this equal to 2:
\[ \sqrt{(x - 1)^2 + (y - \sqrt{3})^2} = 2 \implies (x - 1)^2 + (y - \sqrt{3})^2 = 4 \quad \text{(Equation 2)} \]

### Step 5: Expand Equation 2
Expanding Equation 2:
\[ (x - 1)^2 + (y - \sqrt{3})^2 = 4 \]
\[ (x^2 - 2x + 1) + (y^2 - 2y\sqrt{3} + 3) = 4 \]
Combining terms:
\[ x^2 + y^2 - 2x - 2y\sqrt{3} + 4 = 4 \]
This simplifies to:
\[ x^2 + y^2 - 2x - 2y\sqrt{3} = 0 \]

### Step 6: Substitute Equation 1 into the expanded Equation 2
From Equation 1, we know \(x^2 + y^2 = 4\). Substitute this into the equation:
\[ 4 - 2x - 2y\sqrt{3} = 0 \]
Rearranging gives:
\[ 2x + 2y\sqrt{3} = 4 \implies x + y\sqrt{3} = 2 \quad \text{(Equation 3)} \]

### Step 7: Solve for y in terms of x
From Equation 3:
\[ y\sqrt{3} = 2 - x \implies y = \frac{2 - x}{\sqrt{3}} \]

### Step 8: Substitute y back into Equation 1
Substituting \(y\) into Equation 1:
\[ x^2 + \left(\frac{2 - x}{\sqrt{3}}\right)^2 = 4 \]
Expanding:
\[ x^2 + \frac{(2 - x)^2}{3} = 4 \]
Multiplying through by 3 to eliminate the fraction:
\[ 3x^2 + (2 - x)^2 = 12 \]
Expanding:
\[ 3x^2 + (4 - 4x + x^2) = 12 \]
Combining like terms:
\[ 4x^2 - 4x + 4 = 12 \implies 4x^2 - 4x - 8 = 0 \]
Dividing by 4:
\[ x^2 - x - 2 = 0 \]

### Step 9: Factor the quadratic equation
Factoring gives:
\[ (x - 2)(x + 1) = 0 \]
Thus, \(x = 2\) or \(x = -1\).

### Step 10: Find corresponding y values
1. If \(x = 2\):
\[ y = \frac{2 - 2}{\sqrt{3}} = 0 \implies R = (2, 0) \]

2. If \(x = -1\):
\[ y = \frac{2 - (-1)}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \implies R = (-1, \sqrt{3}) \]

### Conclusion
The possible coordinates of R are (2, 0) and (-1, √3).