Area of a Triangle: Calculating in two ways

Let's solve the problem using the information given:

• We know the area of triangle $ABC$ is given by the formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
• For calculating $AE$, use the entire base $BC$ which is $4 \frac{2}{3} \, \text{cm}$. Convert this to an improper fraction: $BC = \frac{14}{3} \, \text{cm}$.
• Plug into the area formula: $56 = \frac{1}{2} \times \frac{14}{3} \times AE$.
• Solve for $AE$: $AE = \frac{56 \times 2 \times 3}{14} = 24 \, \text{cm}$.
• For calculating $AC$, use base $BD$: $\text{Area} = \frac{1}{2} \times AC \times BD$.
• Substitute known values: $56 = \frac{1}{2} \times AC \times 7$.
• Solve for $AC$: $AC = \frac{56 \times 2}{7} = 16 \, \text{cm}$.

Therefore, the lengths are $AC = 16 \, \text{cm}$ and $AE = 24 \, \text{cm}$.

The correct choice is option 2:
AC = 16
AE = 24

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 $S = 50 \text{cm}^2$.

To solve this problem, we will find the length of side DF using the formula for the area of a triangle:

The area of a triangle is given by:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

For triangle DEF, the area is given as 40 cm², and the height GE is 10 cm. We can consider side DF as the base. Therefore, substitute the given values:
$\frac{1}{2} \times \text{DF} \times 10 = 40$

Simplify this expression:
$\text{DF} \times 5 = 40$

Divide both sides by 5 to solve for DF:
$\text{DF} = \frac{40}{5} = 8$

Thus, the length of side DF is $\text{8 cm}$.

By comparing with the given choices, the correct answer is indeed choice 1, which is 8 cm.

Therefore, the solution to the problem is $\text{8 cm}$.