Area of a Parallelogram: Calculating in two ways

To solve this problem, we'll determine the area of the parallelogram using both given heights and their corresponding bases to verify consistency.

The area of a parallelogram can be calculated using the formula:

$\text{Area} = \text{Base} \times \text{Height}$

First, we calculate the area using $DC$ as the base and $AE$ as the height:

• $\text{Base} = DC = 8 \, \text{cm}$
• $\text{Height} = AE = 3.5 \, \text{cm}$

$\text{Area} = 8 \times 3.5 = 28 \, \text{cm}^2$

Second, we verify the area using $AD$ as the base and $CF$ as the height:

• $\text{Base} = AD = 4 \, \text{cm}$
• $\text{Height} = CF = 7 \, \text{cm}$

$\text{Area} = 4 \times 7 = 28 \, \text{cm}^2$

Since both calculations result in the same area, the solution is consistent.

Therefore, the area of the parallelogram is $28 \, \text{cm}^2$.

In this exercise, we are given two heights and two sides.

It is important to keep in mind: The external height can also be used to calculate the area

Therefore, we can perform the operation of the following exercise:

The height BF * the side AD

8*6

The height BE the side DC
4*12

 The solution of these two exercises is 48, which is the area of the parallelogram.

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

• Step 1: Identify the given dimensions and properties of parallelogram $ABCD$.
• Step 2: Calculate the area of the parallelogram using the known base and height.
• Step 3: Use the area and other dimensions to determine the required segment $AF$.

Now, let's work through each step:

Step 1: Parallelogram $ABCD$ has $AB = 6$ cm and $AD = 4$ cm. The height from $B$ opposite $AD$ is $3$ cm.

Step 2: Calculate the area with base $AB$:

$\text{Area} = 6 \times 3 = 18$ square centimeters.

Step 3: Use base $AD$ to find $AF$ (height):

$18 = 4 \times AF$.

Solve for $AF$:

$AF = \frac{18}{4} = 4.5$ cm.

Therefore, the solution to the problem is $AF = 4.5$ cm.

The problem at hand involves finding the length of side $DC$ using the properties of a parallelogram:

• Step 1: Recognize that in a parallelogram, opposite sides are equal, meaning $AB = DC$ and $AD = BC$.
• Step 2: Analyze the given measurements and diagram. The length labeled as 13 cm does not correspond directly to the parallelogram's sides as vertices are placed differently due to the geometric depiction of parallel lines.
• Step 3: Evaluate the diagram: it's lacking enough information directly connecting the labeled lines to either $AB$, the opposing side length $DC$, or using $AD$ where specified.
• Step 4: Conclusion based on information presented: no direct measure or calculative method allows adherence to standard formulas or express calculation of $DC$ based on what is explicitly visualized and labeled.

Given the insufficient data to deduce the length of $DC$ through standard parallelogram properties, the resolution is that it is indeed impossible to determine $DC$ using provided labels and geometric read from the diagram alone.

Therefore, the correct conclusion is: It is not possible to calculate.

To solve for the length of $DF$, let's consider both ways of calculating the area of parallelogram $ABCD$:

• Step 1: Calculate area using base $AB$:
  $\text{Area} = AB \times ED = 12 \times 8 = 96$ square cm.
• Step 2: Calculate area using base $BC$:
  $\text{Area} = BC \times DF = 10 \times DF$.
• Equate 96 to $10 \times DF$:
  $10 \times DF = 96$.
• Step 3: Solve for $DF$ by dividing by 10:
  $DF = \frac{96}{10} = 9.6$ cm.

Therefore, the length of $DF$ is $9.6$ cm.

Hence, the correct answer is choice 4, which is $9.6$ cm.