To calculate the area of quadrilateral ABCD where:

• ∠A = 90°
• AB = 32 cm
• AD = 24 cm
• BC = CD = 25 cm

We can break the quadrilateral into two triangles: △ABD and △BCD.

Step 1: Area of triangle ABD

Since ∠A = 90°, triangle ABD is a right-angled triangle. The area of a right-angled triangle can be calculated as:

$\text{Area of } \triangle ABD = \frac{1}{2} \times AB \times AD$

Substituting the given values:

$\text{Area of } \triangle ABD = \frac{1}{2} \times 32 \times 24 = 384 \, \text{cm}^2$

Step 2: Area of triangle BCD

For △BCD, we use Heron's formula since it's not a right-angled triangle. First, find the semi-perimeter $s$ of △BCD:

$s = \frac{BC + CD + BD}{2}$

We don't know $BD$ yet, so we will first calculate it using the Pythagorean theorem in triangle ABD.

$BD = \sqrt{AB^2 + AD^2} = \sqrt{32^2 + 24^2} = \sqrt{1024 + 576} = \sqrt{1600} = 40 \, \text{cm}$

Now that we know $BD = 40$ cm, we can calculate the semi-perimeter $s$ of △BCD:

$s = \frac{25 + 25 + 40}{2} = 45 \, \text{cm}$

Next, apply Heron's formula to find the area of △BCD:

$\text{Area of } \triangle BCD = \sqrt{s(s - BC)(s - CD)(s - BD)}$

Substitute the values:

$\text{Area of } \triangle BCD = \sqrt{45(45 - 25)(45 - 25)(45 - 40)}$ $= \sqrt{45 \times 20 \times 20 \times 5} = \sqrt{90000} = 300 \, \text{cm}^2$

Step 3: Total area of quadrilateral ABCD

Now, sum the areas of △ABD and △BCD to get the total area of the quadrilateral ABCD:

$\text{Total area} = 384 \, \text{cm}^2 + 300 \, \text{cm}^2 = 684 \, \text{cm}^2$

Thus, the area of the quadrilateral ABCD is 684 cm².

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Would you like more details on any of these steps or have further questions? Here are some related questions for further exploration:

1. How does Heron's formula apply to non-right-angled triangles?
2. What other methods can be used to calculate the area of irregular quadrilaterals?
3. How do the properties of right-angled triangles simplify area calculations?
4. What is the importance of the Pythagorean theorem in geometry?
5. How can you calculate the area of a quadrilateral with no right angles?

Tip: Always double-check if a shape can be broken down into simpler geometric figures when calculating area.