To find the area of quadrilateral ABCD, let's follow these steps:

Step 1: Understanding the Problem

We have a quadrilateral ABCD with:

• A perimeter of 60 cm.
• AB = 9 cm.
• Sides AD and BC are parallel with a distance of 7 cm between them.

Points E and F are chosen on sides BC and AD, respectively, such that the line EF divides ABCD into two quadrilaterals with equal perimeters and equal areas.

Step 2: Define Unknowns

Let's denote the lengths of the sides as follows:

• $AB = 9$ cm (given).
• $AD = x$ cm.
• $BC = y$ cm.
• $CD = z$ cm.

The total perimeter is given by: $AB + BC + CD + AD = 9 + x + y + z = 60 \text{ cm}.$ Thus, we have: $x + y + z = 51 \text{ cm} \quad \text{(Equation 1)}.$

Step 3: Use the Property of Equal Areas and Perimeters

The line EF divides the quadrilateral ABCD into two smaller quadrilaterals with equal perimeters and equal areas. Because AD and BC are parallel with a distance of 7 cm, these two smaller quadrilaterals will be trapezoids sharing the same height, i.e., 7 cm.

Since the perimeters and areas of the two smaller quadrilaterals are equal, it implies that the bases of these trapezoids are equal: $x + y = z + 9 \quad \text{(Equation 2)}.$

From Equation 1: $z = 51 - (x + y).$ Substituting Equation 2 into this: $z = 51 - (z + 9).$ Simplify this equation: $2z + 9 = 51.$ $2z = 42.$ $z = 21 \text{ cm}.$

Now, substituting $z = 21$ cm into Equation 1: $x + y + 21 = 51.$ $x + y = 30 \text{ cm}.$

Using Equation 2: $x + y = z + 9 \implies 30 = 21 + 9.$ This confirms that our values for $x, y,$ and $z$ are consistent with the conditions given.

Step 4: Calculate the Area of Quadrilateral ABCD

Since AD and BC are parallel with a height of 7 cm, ABCD is a trapezoid. The area $A$ of a trapezoid is given by: $A = \frac{1}{2} \times (\text{Sum of the parallel sides}) \times \text{Height}.$ Substituting the values: $A = \frac{1}{2} \times (AD + BC) \times 7 = \frac{1}{2} \times (30) \times 7 = 105 \text{ cm}^2.$

Thus, the area of quadrilateral ABCD is $\boxed{105 \text{ cm}^2}$.

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Would you like more details on this solution or have any questions?

Here are 5 related questions you might find interesting:

1. How do you find the area of a quadrilateral when only its sides are known?
2. What are the conditions for a quadrilateral to be a trapezoid?
3. How does the area formula change if the height of the trapezoid is doubled?
4. What is the effect on the area if the perimeter of the quadrilateral changes but the height remains the same?
5. How would you find the area if the quadrilateral were not a trapezoid?

Tip: When dealing with problems involving equal divisions by lines within a shape, always look for properties of symmetry and equal lengths that can simplify your calculations.