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

Question

Shown below is the deltoid ABCD.

Side length BD equals 5 cm.

The area of the deltoid is 45 cm².

What is the the value of $a?$

00:00 Find A

00:03 The entire side equals the sum of its parts

00:11 The size of diagonal BD according to the given data

00:15 We'll use the formula for calculating the area of a kite

00:18 (diagonal times diagonal) divided by 2

00:21 We'll substitute appropriate values according to the given data and solve for A

00:31 We'll isolate A

00:45 And this is the solution to the question

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

We are given a deltoid ABCD where:

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 $4.5 \, \text{cm}$ .

$4.5$