Find the area (correct to three significant digits) of quadrilateral ABCD with angle BCA = 90°, AB = 26 cm and ACD as an equilateral triangle of side 24 cm.

Answer

To find the area of quadrilateral ABCD, we will break it down into two parts: the area of triangle ACD and the area of triangle ABC.

### Step 1: Calculate the area of triangle ACD
Since triangle ACD is an equilateral triangle with each side measuring 24 cm, 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 the value of \( a \):

\[ \text{Area}_{ACD} = \frac{\sqrt{3}}{4} \times 24^2 \]

Calculating \( 24^2 \):

\[ 24^2 = 576 \]

Now substituting back into the area formula:

\[ \text{Area}_{ACD} = \frac{\sqrt{3}}{4} \times 576 \]

Using \( \sqrt{3} \approx 1.732 \):

\[ \text{Area}_{ACD} \approx \frac{1.732}{4} \times 576 \approx 249.00 \text{ cm}^2 \]

Rounding to three significant digits gives us:

\[ \text{Area}_{ACD} \approx 249 \text{ cm}^2 \]

### Step 2: Calculate the area of triangle ABC
Triangle ABC is a right triangle with angle BCA = 90°. We know the lengths of AB and AC.

- \( AB = 26 \) cm
- \( AC = 24 \) cm

To find the height (BC), we will use the Pythagorean theorem:

\[ AB^2 = AC^2 + BC^2 \]

Substituting the known values:

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

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

\[ 676 = 576 + BC^2 \]

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

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

Taking the square root:

\[ BC = 10 \text{ cm} \]

Now we can calculate the area of triangle ABC using the formula for the area of a triangle:

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

Here, the base is AC (24 cm) and the height is BC (10 cm):

\[ \text{Area}_{ABC} = \frac{1}{2} \times 24 \times 10 = 120 \text{ cm}^2 \]

### Step 3: Calculate the total area of quadrilateral ABCD
Now, we can find the total area of quadrilateral ABCD by adding the areas of triangles ACD and ABC:

\[ \text{Area}_{ABCD} = \text{Area}_{ACD} + \text{Area}_{ABC} \]

Substituting the values:

\[ \text{Area}_{ABCD} = 249 + 120 = 369 \text{ cm}^2 \]

Thus, the area of quadrilateral ABCD is:

\[ \text{Area}_{ABCD} = 369 \text{ cm}^2 \]

### Final Answer
The area of quadrilateral ABCD is \( 369 \text{ cm}^2 \).

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