Deltoid Geometry: Find Variable b When Area = 144 cm²

To solve the problem, we'll follow the steps outlined:

• Step 1: Identify the given information.
• Step 2: Apply the appropriate formula for the area of the deltoid.
• Step 3: Solve for $b$ in terms of known values.

Let's break it down:

Step 1: We know the length of diagonal $BM = 4$ cm and the area of the deltoid is $144$ cm².

Step 2: The area of a deltoid is given by the formula:

$\text{Area} = \frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2$

Here, the diagonals correspond to line segments of the form $2b$ and $4b$ as represented in the setup of the problem.

Step 3: Substituting the values into the area formula, we have:

$\frac{1}{2} \times (2b) \times (4b) = 144$

Simplifying this, we get:

$4b^2 = 144 \quad \Rightarrow \quad b^2 = \frac{144}{4} = 36$

Therefore, solving for $b$, we find:

$b = \sqrt{36} = 6$

Thus, the value of $b$ is $6$.