Deltoid Diagonal Problem: Finding Secondary Diagonal Length with 7:1 Ratio

To solve this problem, we'll follow these steps:

• Step 1: Identify the segment lengths of the main diagonal using the ratio.
• Step 2: Apply the triangle area formula using the secondary diagonal as the base.
• Step 3: Solve for the length of the secondary diagonal.

Step 1: The main diagonal $AD = 40$ cm is divided into segments: $AO$ and $OD$ with a ratio 7:1. Therefore, $AO = \frac{7}{8} \times 40 = 35$ cm, and $OD = \frac{1}{8} \times 40 = 5$ cm.

Step 2: Consider the isosceles triangle $\triangle ABD$ with base $BC$ (the secondary diagonal) and $AB = AD = 40$ cm divided by the diagonals. The area of $\triangle ABD = 20$ cm².

Using the formula for the area of a triangle–$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$–the height is the segment perpendicular to $BC$, which is $AO = 5$ cm.

Thus, $20 = \frac{1}{2} \times BC \times 35$.

Solving for $BC$, we get $BC = \frac{40}{35} = \frac{8}{7} \times 5 = 8$ cm.

Therefore, the length of the secondary diagonal is $8$ cm.