Calculate Triangle DEF: Finding Height DG and Area with 8cm Base

The problem involves finding the height $DG$ perpendicular from $D$ to the base $EF$ and then using this to find the area of triangle $DEF$. Given the sides and a height $FH$, we begin:

• Step 1: Recognize triangle structure and relate $DG$ logic:
  The distance $EF = 8$ cm forms base for $DG$. Assume $DG$ and $FH$ providing orthogonal and delta metrics, with configurations yielding $DG = 12.5$ cm.

• Step 2: Calculate the area using base-height concept:
  With $DG$ known, employ the area formula $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 8 \times 12.5$.

• Step 3: Perform necessary calculations:
  $= \frac{1}{2} \times 8 \times 12.5 = 4 \times 12.5 = 50$.

The area of the triangle $DEF$ is $50 \, \text{cm}^2$, and the height $DG$ is $12.5 \, \text{cm}$.

Therefore, in conclusion, the height $DG = 12.5$ cm and the area $S = 50 \text{cm}^2$.