Deltoid Area Problem: Find Variable 'a' Given Area 45 cm²

Let's solve this problem by working through the steps:

We are given a deltoid ABCD where:

• The length of diagonal BD is 5 cm.
• The sum of the segments forming diagonal AC is $a + 3a = 4a$.
• The area of the deltoid is 45 cm².

We use the area formula for a deltoid when the diagonals intersect at right angles:

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

Here, $d_1 = 5$ cm and $d_2 = 4a$. Substituting these values into the formula:

$\frac{1}{2} \times 5 \times 4a = 45$

Simplifying this equation:

$\frac{1}{2} \times 20a = 45$

$10a = 45$

Now, solve for $a$:

$a = \frac{45}{10}$

$a = 4.5$

Therefore, the length of segment $a$ is $4.5 \, \text{cm}$.