Q2. Calculate the area of a quadrilateral ABCD in which angle A = 900, AB = 32cm, AD = 24cm and BC = CD = 25cm. 

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**.

---

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.

Q2. Calculate the area of a quadrilateral ABCD in which angle A = 90°, AB = 32 cm, AD = 24 cm and BC = CD = 25 cm.

Learn how to calculate the area of quadrilateral ABCD, where angle A = 90°, AB = 32 cm, AD = 24 cm, and BC = CD = 25 cm. This solution involves breaking the quadrilateral into two triangles and applying both the Pythagorean Theorem and Heron's Formula. Ideal for Grades 9-12.

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