Area of a Deltoid: Using external height

To solve the problem of finding the area of the deltoid (kite) ABCD, we will follow these steps:

• Step 1: Identify the given diagonal lengths. Here, $AC = 5$ cm and $BD = 8$ cm.
• Step 2: Use the formula for the area of a kite or deltoid: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ where $d_1$ and $d_2$ are the lengths of the diagonals.
• Step 3: Plug in the given values into the formula to calculate the area.

Now, let's calculate:
- The length of diagonal $AC = 5$ cm.
- The length of diagonal $BD = 8$ cm.

Applying the formula:

$\text{Area} = \frac{1}{2} \times 5 \times 8 = \frac{1}{2} \times 40 = 20$

Therefore, the area of the deltoid is $20$ cm².

To find the area of the deltoid ABCD, we use the external height formula for deltoids:

Given:
- Height ($h$) = $16$ cm
- Segment related to base ($b$) = $5$ cm

The area of the deltoid can be calculated by:

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

Plugging in our values, we have:

$\text{Area} = \frac{1}{2} \times 5 \times 16$

Calculating the result:

$\text{Area} = \frac{1}{2} \times 80 = 40$ cm$^2$

Therefore, the area of deltoid ABCD is $40$ cm$^2$.

We are tasked with finding the area of the deltoid (or kite) ABCD using the lengths of its diagonals. The given diagonals are $AC = 5$ cm and $BD = 18$ cm. The diagonals of a kite are perpendicular to each other.

To find the area of the kite, we use the formula:

$\text{Area} = \frac{1}{2} \times d_1 \times d_2$

Substituting the given values ($d_1 = 5$ cm and $d_2 = 18$ cm) into the formula, we get:

$\text{Area} = \frac{1}{2} \times 5 \times 18 = \frac{1}{2} \times 90 = 45 \text{ cm}^2$

Hence, the area of the deltoid ABCD is $45$ cm².

To solve the problem of finding the area of the deltoid $ABCD$, we will use the area formula for a kite. The formula is:

$\text{Area} = \frac{1}{2} \times d_1 \times d_2$

Given:

• $d_1 = 22$ cm (one diagonal of the deltoid)
• $d_2 = 5$ cm (the other diagonal of the deltoid)

Substitute the given values into the formula:

$\text{Area} = \frac{1}{2} \times 22 \times 5$

$\text{Area} = \frac{1}{2} \times 110$

$\text{Area} = 55 \text{ cm}^2$

Therefore, the area of the deltoid $ABCD$ is $\boxed{55}$ square centimeters.

To find the area of a deltoid (also known as a kite), we need to make use of the given dimensions: the kite's longer diagonal ($AC$) and the shorter diagonal ($BD$). Here are the steps we'll follow:

• Step 1: Identify the key information.
• Step 2: Use the formula for the area of a kite.
• Step 3: Perform the calculation.

Let's go through each step in detail:

Step 1: Identify the key information
In the problem, the deltoid (kite) is described with vertices $A$, $B$, $C$, and $D$. From the diagram, we have the following measurements:

• The diagonal $AC = 19$ cm.
• The diagonal $BD = 5$ cm.

Step 2: Use the formula for the area of a kite
The area of a kite can be calculated using the formula: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ where $d_1$ and $d_2$ are the lengths of the diagonals.

Step 3: Perform the calculation
Now we substitute the given measurements into the formula:

$\text{Area} = \frac{1}{2} \times 19 \times 5$

Carrying out the multiplication:

$\text{Area} = \frac{1}{2} \times 95 = 47.5$

Thus, the area of the deltoid (kite) is $47.5$ cm².

This matches choice $\mathbf{3}$.