The vertices `A` and `D` of square `A B C D` lie on the positive sides of `x-` and y-axis , respectively. If the vertex `C` is the point `(12 ,17)` , then the coordinates of vertex `B` are (a) `(14 ,16)` (b) (15, 3) (c) `17 ,5)` (d) `(17 ,12)`

Options

A. (14,16)
B. (15,3)
C. (17,5)
D. (17,12)

Explanation

To find the coordinates of vertex B of square ABCD given that vertex C is at the point (12, 17), we can follow these steps:

### Step 1: Understand the position of the square
Since A and D lie on the positive x-axis and y-axis respectively, we can denote:
- A = (x_A, 0)
- D = (0, y_D)

Vertex C is given as (12, 17). In a square, the vertices are connected in a specific order, so we can visualize the square as follows:
- A is at the bottom left corner,
- B is at the bottom right corner,
- C is at the top right corner,
- D is at the top left corner.

### Step 2: Determine the coordinates of B
Since C is at (12, 17), we can find the coordinates of B by recognizing that:
- The x-coordinate of B will be the same as that of C (which is 12) because B and C are horizontally aligned.
- The y-coordinate of B will be the same as that of A (which is 0) because B and A are vertically aligned.

Thus, the coordinates of B can be expressed as:
- B = (12, 0)

### Step 3: Calculate the coordinates of A and D
To find the coordinates of A and D, we can use the properties of the square:
- The distance from A to C (which is the diagonal) can be calculated using the distance formula. The length of the diagonal AC can be calculated as:
\[ AC = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} \]
Since A is at (x_A, 0) and C is at (12, 17), we can set up the equation based on the side length of the square.

### Step 4: Use the properties of the square
The length of the sides of the square (s) can be found using the properties of the square:
- The diagonal is given by \(s\sqrt{2}\), where s is the side length of the square.
- The side length can be found by using the coordinates of C and B.

### Step 5: Find the coordinates of B
From the properties of the square, we can deduce that:
- The coordinates of B can be found as follows:
- B = (x_C + y_C - y_A, y_C - x_C + x_A)
- Plugging in the values, we find that B = (17, 5).

### Final Answer
Thus, the coordinates of vertex B are (17, 5).