Deltoid Diagonal Problem: Finding Triangle Ratios with 28cm and 13cm Measurements

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

• Step 1: Use the given ratio to find the lengths AP and PC.
• Step 2: Calculate the areas of triangles ABD and CBD using their respective heights.
• Step 3: Determine the area ratio of the two triangles.

Now, let's work through each step:

Step 1: Divide Main Diagonal (AC).
The main diagonal AC is 28 cm long and is divided by the secondary diagonal in the ratio 4:3. Therefore, the segments AP and PC can be found using inversion proportionality:

• AP = $28 \times \frac{4}{7} = 16 \, \text{cm}$
• PC = $28 \times \frac{3}{7} = 12 \, \text{cm}$

Step 2: Calculate Areas of Triangles ABD and CBD.
Triangles ABD and CBD each have the common base, BD = 13 cm. Given the symmetry:

• Area of triangle ABD = $\frac{1}{2} \times 13 \times 16 \, \text{cm}^2 = 104 \, \text{cm}^2$
• Area of triangle CBD = $\frac{1}{2} \times 13 \times 12 \, \text{cm}^2 = 78 \, \text{cm}^2$

Step 3: Determine the Ratio of Areas.
The ratio of the areas of triangle ABD to triangle CBD is: $\frac{104}{78} = \frac{52}{39} = \frac{4}{3}$

Therefore, the solution to the problem is $4:3$.