Calculate the area of quadrilateral ABCD, in which `angleABD=90^(@)`, triangle BCD is an equilateral triangle of side 24 cm and AD = 26 cm.

Answer

To calculate the area of quadrilateral ABCD, we will break it down into two parts: the area of triangle BCD and the area of triangle ABD.

### Step 1: Understand the given information
- Triangle BCD is an equilateral triangle with each side measuring 24 cm.
- AD = 26 cm.
- Angle ABD = 90°.

### Step 2: Calculate the area of triangle BCD
Since BCD is an equilateral triangle, we can use the formula for the area of an equilateral triangle:

\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]

where \( a \) is the length of a side.

Substituting \( a = 24 \) cm:

\[ \text{Area of BCD} = \frac{\sqrt{3}}{4} \times (24)^2 \]

Calculating \( (24)^2 \):

\[ (24)^2 = 576 \]

Now substituting back into the area formula:

\[ \text{Area of BCD} = \frac{\sqrt{3}}{4} \times 576 = \frac{576\sqrt{3}}{4} = 144\sqrt{3} \approx 249.43 \text{ cm}^2 \]

### Step 3: Calculate the area of triangle ABD
Triangle ABD is a right triangle. The area of a right triangle can be calculated using the formula:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

In triangle ABD:
- The height (BD) = 24 cm (since BCD is equilateral).
- The base (AB) is unknown, so we need to find it using the Pythagorean theorem.

### Step 4: Use the Pythagorean theorem to find AB
In triangle ABD, we have:

\[ AD^2 = AB^2 + BD^2 \]

Substituting the known values:

\[ 26^2 = AB^2 + 24^2 \]

Calculating \( 26^2 \) and \( 24^2 \):

\[ 676 = AB^2 + 576 \]

Now, solving for \( AB^2 \):

\[ AB^2 = 676 - 576 = 100 \]

Taking the square root:

\[ AB = \sqrt{100} = 10 \text{ cm} \]

### Step 5: Calculate the area of triangle ABD
Now we can calculate the area of triangle ABD:

\[ \text{Area of ABD} = \frac{1}{2} \times AB \times BD = \frac{1}{2} \times 10 \times 24 \]

Calculating:

\[ \text{Area of ABD} = \frac{1}{2} \times 240 = 120 \text{ cm}^2 \]

### Step 6: Calculate the total area of quadrilateral ABCD
Now we can find the total area of quadrilateral ABCD by adding the areas of triangles BCD and ABD:

\[ \text{Total Area} = \text{Area of BCD} + \text{Area of ABD} \]

Substituting the values:

\[ \text{Total Area} = 249.43 + 120 = 369.43 \text{ cm}^2 \]

### Final Answer
The area of quadrilateral ABCD is approximately **369.43 cm²**.

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