Problem:

Given a square ABCD with a point P inside such that PA = 3, PB = 7, and PD = 5. Find the area of square ABCD.

Solution:

To find the area of square ABCD, we need to determine the length of its sides. Let's break down the problem step-by-step.

Step 1: Identify the properties of the given square ABCD.

- ABCD is a square, which means all its sides are equal in length.
- Let's assume the length of each side of square ABCD as 'x'.

Step 2: Analyze the given information.

- PA = 3
- PB = 7
- PD = 5

Step 3: Use the given information to form equations.

- In square ABCD, PA = PD = 5 (opposite sides of a square are equal).
- In square ABCD, PB is the diagonal of the square.

Step 4: Apply the Pythagorean theorem to find the length of the sides.

- Using the Pythagorean theorem, we know that the diagonal of a square is √(2 * side^2).
- Therefore, PB = √(2 * x^2).
- Substituting the value of PB as 7, we have 7 = √(2 * x^2).

Step 5: Solve the equation to find the value of x.

- Squaring both sides of the equation, we get 49 = 2 * x^2.
- Simplifying further, we have x^2 = 24.5.
- Taking the square root of both sides, we find x = √24.5.

Step 6: Calculate the area of square ABCD.

- The area of a square is given by the formula A = side^2.
- Substituting the value of x (√24.5) in the formula, the area of square ABCD is A = (√24.5)^2 = 24.5.

Therefore, the area of square ABCD is 24.5 square units.