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The question asks to:

1. **Construct a quadrilateral ABCD** where:
   - \( AB = 4.5 \, \text{cm} \)
   - \( AC = CD = 5 \, \text{cm} \)
   - \( AD = 6 \, \text{cm} \)
   - \( \angle BAC = 60^\circ \)

2. **Construct a triangle PBC** such that its area is equal to the area of the quadrilateral ABCD.

For part **(b)**, you need to **explain why** the area of quadrilateral ABCD is equal to the area of triangle PBC.

### Plan:

1. **Step 1 (Quadrilateral ABCD Construction):**
   - Draw \( AB = 4.5 \, \text{cm} \).
   - At point A, construct \( \angle BAC = 60^\circ \).
   - Measure and mark \( AC = 5 \, \text{cm} \).
   - With point C as the center and radius \( CD = 5 \, \text{cm} \), draw an arc.
   - With point A as the center and radius \( AD = 6 \, \text{cm} \), draw another arc intersecting the previous arc at point D.
   - Join points \( C \) and \( D \) to complete the quadrilateral.

2. **Step 2 (Triangle PBC Construction):**
   - Use the area formula for quadrilateral ABCD (which can be calculated geometrically or using Heron's formula for triangles).
   - The area of triangle \( PBC \) should be made equal to that of quadrilateral \( ABCD \).
   - You can adjust point \( P \) accordingly to meet this area condition.

Would you like a detailed geometric or algebraic solution for calculating the areas and completing the constructions?

Here are 5 related questions to expand the discussion:
1. How can the area of quadrilateral ABCD be calculated using coordinates or geometric properties?
2. How does the construction of triangle PBC ensure the areas are equal?
3. What are the key geometric properties of a triangle with equal area to a given quadrilateral?
4. How does the angle \( \angle BAC = 60^\circ \) impact the construction of the quadrilateral?
5. How can we verify if two areas are exactly equal using geometric methods?

**Tip:** When constructing quadrilaterals, use precise geometric tools to ensure correct angles and side lengths for accurate area calculations.

Construct a quadrilateral ABCD where AB = 4.5 cm, AC = CD = 5 cm, AD = 6 cm, and ∠BAC = 60°. Also, construct a triangle PBC whose area is equal to the area of the quadrilateral. Additionally, explain why the areas of quadrilateral ABCD and triangle PBC are equal.

Learn how to construct a quadrilateral ABCD and a triangle PBC where the areas are equal. This problem involves geometric constructions, area calculation techniques, and reasoning for equal areas between the two shapes. Suitable for students in Grades 9-11.

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