Which of the following points lies the greatest distance from the origin in the xy-plane?

Options

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

Answer

To determine which of the given points lies the greatest distance from the origin in the xy-plane, we will use the distance formula. The distance \( d \) from a point \( (x, y) \) to the origin \( (0, 0) \) is given by:

\[ d = \sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2} \]

Now, we will calculate the distance for each point provided.

### Step 1: Calculate the distance for the first point \((- \frac{3}{2}, - \frac{3}{2})\)

\[ d_1 = \sqrt{\left(-\frac{3}{2}\right)^2 + \left(-\frac{3}{2}\right)^2} \]
\[ = \sqrt{\frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{18}{4}} = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}} \]

### Step 2: Calculate the distance for the second point \((-1, -1)\)

\[ d_2 = \sqrt{(-1)^2 + (-1)^2} \]
\[ = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the distance for the third point \((- \frac{1}{2}, 0)\)

\[ d_3 = \sqrt{\left(-\frac{1}{2}\right)^2 + 0^2} \]
\[ = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Step 4: Calculate the distance for the fourth point \((0, 1)\)

\[ d_4 = \sqrt{0^2 + 1^2} \]
\[ = \sqrt{1} = 1 \]

### Step 5: Compare the distances

Now we have the distances:
- \( d_1 = \frac{3}{\sqrt{2}} \)
- \( d_2 = \sqrt{2} \)
- \( d_3 = \frac{1}{2} \)
- \( d_4 = 1 \)

To compare these distances, we can convert them to a common format or approximate their values:
- \( d_1 \approx 2.12 \)
- \( d_2 \approx 1.41 \)
- \( d_3 = 0.5 \)
- \( d_4 = 1 \)

### Conclusion

The greatest distance from the origin is \( d_1 = \frac{3}{\sqrt{2}} \). Therefore, the point that lies the greatest distance from the origin is:

**Answer: \((- \frac{3}{2}, - \frac{3}{2})\)**

---