Segments AP and BP have the same length. If the coordinates of A and P are `(-1, 0) and (4, 12)`, respectively, which could be the coorinate of B ?
I. `((3)/(2), 6)`
II. `(9, 24)`
III. `(-8, 7)`

Options

A. I and II only
B. II and III only
C. II only
D. III only

Answer

To solve the problem, we need to determine which of the given coordinates for point B results in the same distance from point P as the distance from point A to point P.

### Step 1: Calculate the distance between points A and P

Given:
- A = (-1, 0)
- P = (4, 12)

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of A and P:
\[ d_{AP} = \sqrt{(4 - (-1))^2 + (12 - 0)^2} \]
\[ = \sqrt{(4 + 1)^2 + (12 - 0)^2} \]
\[ = \sqrt{5^2 + 12^2} \]
\[ = \sqrt{25 + 144} \]
\[ = \sqrt{169} \]
\[ = 13 \]

### Step 2: Check the distance from P to each of the given coordinates for B

#### Option I: B = \((\frac{3}{2}, 6)\)

Calculate the distance \(d_{BP}\):
\[ d_{BP} = \sqrt{(4 - \frac{3}{2})^2 + (12 - 6)^2} \]
\[ = \sqrt{(\frac{8}{2} - \frac{3}{2})^2 + (6)^2} \]
\[ = \sqrt{(\frac{5}{2})^2 + 6^2} \]
\[ = \sqrt{\frac{25}{4} + 36} \]
\[ = \sqrt{\frac{25}{4} + \frac{144}{4}} \]
\[ = \sqrt{\frac{169}{4}} \]
\[ = \frac{13}{2} \]

Since \(\frac{13}{2} \neq 13\), option I is **not valid**.

#### Option II: B = (9, 24)

Calculate the distance \(d_{BP}\):
\[ d_{BP} = \sqrt{(4 - 9)^2 + (12 - 24)^2} \]
\[ = \sqrt{(-5)^2 + (-12)^2} \]
\[ = \sqrt{25 + 144} \]
\[ = \sqrt{169} \]
\[ = 13 \]

Since \(d_{BP} = 13\), option II is **valid**.

#### Option III: B = (-8, 7)

Calculate the distance \(d_{BP}\):
\[ d_{BP} = \sqrt{(4 - (-8))^2 + (12 - 7)^2} \]
\[ = \sqrt{(4 + 8)^2 + (12 - 7)^2} \]
\[ = \sqrt{(12)^2 + (5)^2} \]
\[ = \sqrt{144 + 25} \]
\[ = \sqrt{169} \]
\[ = 13 \]

Since \(d_{BP} = 13\), option III is **valid**.

### Conclusion

The valid coordinates for point B are:
- Option II: (9, 24)
- Option III: (-8, 7)