In parallelogram ABCD, AB = 10 cm. The altitudes corresponding to the sides AB and AD are respectively 7 cm and 8 cm. If AD is k cm. Then value of 4 k is

Answer

To solve the problem, we need to find the value of \(4k\) where \(k\) is the length of side \(AD\) in parallelogram \(ABCD\). We are given the following information:

- \(AB = 10 \, \text{cm}\)
- The altitude corresponding to side \(AB\) (denoted as \(DF\)) is \(7 \, \text{cm}\).
- The altitude corresponding to side \(AD\) (denoted as \(BE\)) is \(8 \, \text{cm}\).
- \(AD = k \, \text{cm}\).

### Step-by-Step Solution:

1. **Write the formula for the area of the parallelogram**:
The area of a parallelogram can be calculated using the formula:
\[ \text{Area} = \text{Base} \times \text{Height} \]

2. **Calculate the area using base \(AB\)**:
Using \(AB\) as the base:
\[ \text{Area} = AB \times DF = 10 \, \text{cm} \times 7 \, \text{cm} = 70 \, \text{cm}^2 \]

3. **Calculate the area using base \(AD\)**:
Using \(AD\) as the base:
\[ \text{Area} = AD \times BE = k \, \text{cm} \times 8 \, \text{cm} = 8k \, \text{cm}^2 \]

4. **Set the two area equations equal to each other**:
Since both expressions represent the area of the same parallelogram, we can set them equal:
\[ 70 = 8k \]

5. **Solve for \(k\)**:
To find \(k\), divide both sides by \(8\):
\[ k = \frac{70}{8} = \frac{35}{4} \, \text{cm} \]

6. **Calculate \(4k\)**:
Now, we need to find \(4k\):
\[ 4k = 4 \times \frac{35}{4} = 35 \]

### Final Answer:
The value of \(4k\) is \(35\).

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