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The question in the image involves the application of quadratic equations to solve a geometry problem. Here's a breakdown of the problem:

### Problem Description
- **Given:**
  - A quadrilateral \(ABCD\) where \(AB = 10\), \(BC = 13\), \(CD = 14\), and \(DA = 7\).
  - Diagonals \(AC\) and \(BD\) intersect at point \(P\).
  - The area of triangle \(APD = 21\) and the area of triangle \(BPC = 63\).

#### (a) Find the area of triangle \(BPD\) if \(BD^2 = 178 + 84\sqrt{2}\).

#### (b) Show that the area of quadrilateral \(ABCD = 84 + 42\sqrt{2}\).

### Solution Approach

Let's start by solving part (a).

1. **Find the Area of Triangle \(BPD\):**

   Given that the diagonals intersect at point \(P\), we know that the sum of the areas of the triangles formed by these diagonals is equal to the area of the quadrilateral.

   \[
   \text{Area of Quadrilateral } ABCD = \text{Area of } APD + \text{Area of } BPD + \text{Area of } BPC.
   \]

2. **Solve for Part (a):**
   
   - Use the properties of diagonals and the information given to find the area of triangle \(BPD\).
   - Then, apply the relationship to calculate the area of quadrilateral \(ABCD\) in part (b).

Would you like me to proceed with a detailed solution for part (a) and (b)? 

### Related Questions

1. What are the properties of a quadrilateral that allow diagonals to intersect at a point?
2. How do you calculate the area of a quadrilateral given its diagonal lengths and intersecting angles?
3. Can you derive the expression for the area of triangle \(BPD\) using only the lengths given?
4. How does the Pythagorean theorem apply to the diagonals intersecting at a point in a quadrilateral?
5. What other geometric properties can be derived if we know the areas of individual triangles within a quadrilateral?

### Tip:
Remember, the properties of triangles within a quadrilateral often give clues to finding areas and relationships between side lengths.

In the figure, ABCD is a rectangle where AB = 13 cm and BC = 5 cm. Diagonals AC and BD meet at P. Show that AC^2 + BD^2 = 2(AB^2 + BC^2). Then, using this, prove that the sum of areas of triangles ABD and ABC is equal to the area of the rectangle.

Explore how to solve a geometry problem involving a rectangle with given side lengths and diagonals. Learn to apply the Pythagorean theorem and quadratic equations to prove the relationship between diagonal lengths and triangle areas within the rectangle. Suitable for Grades 8-10.

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