ABCD is a parallelogram with sides AB = 12 cm, BC= 10 cm and diagonal AC = 16 cm. The area of the parallelogram and the distance between the shorter sides are respectively.

Options

A. `120.5 cm^2, 12.05 cm`
B. `119.8 cm^2 , 11.98 cm`
C. `118.71 cm^2, 11.87 cm`
D. `117.9 cm^2 , 11.79 cm`

Answer

To find the area of the parallelogram ABCD and the distance between the shorter sides, we can follow these steps:

### Step 1: Identify the given values
- Side AB (base) = 12 cm
- Side BC (height) = 10 cm
- Diagonal AC = 16 cm

### Step 2: Calculate the semi-perimeter (s) of triangle ABC
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{AB + BC + AC}{2} \]
Substituting the values:
\[ s = \frac{12 + 10 + 16}{2} = \frac{38}{2} = 19 \text{ cm} \]

### Step 3: Use Heron's formula to find the area of triangle ABC
Heron's formula for the area \( A \) of a triangle is:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Where \( a = AB = 12 \) cm, \( b = BC = 10 \) cm, and \( c = AC = 16 \) cm.

Substituting the values:
\[ A = \sqrt{19(19-12)(19-10)(19-16)} = \sqrt{19 \times 7 \times 9 \times 3} \]

Calculating the individual terms:
- \( 19 - 12 = 7 \)
- \( 19 - 10 = 9 \)
- \( 19 - 16 = 3 \)

Now, calculate:
\[ A = \sqrt{19 \times 7 \times 9 \times 3} \]

Calculating \( 19 \times 7 = 133 \) and \( 9 \times 3 = 27 \):
\[ A = \sqrt{133 \times 27} \]

Calculating \( 133 \times 27 \):
\[ 133 \times 27 = 3591 \]
Thus,
\[ A = \sqrt{3591} \approx 59.9 \text{ cm}^2 \]

### Step 4: Calculate the area of the parallelogram
The area of the parallelogram is twice the area of triangle ABC:
\[ \text{Area of parallelogram} = 2 \times A \approx 2 \times 59.9 \approx 119.8 \text{ cm}^2 \]

### Step 5: Calculate the distance between the shorter sides
The distance (height) between the shorter sides can be calculated using the formula:
\[ \text{Area} = \text{Base} \times \text{Height} \]
Using base \( BC = 10 \) cm:
\[ 119.8 = 10 \times \text{Height} \]
Solving for height:
\[ \text{Height} = \frac{119.8}{10} \approx 11.98 \text{ cm} \]

### Final Results
- Area of the parallelogram: \( 119.8 \text{ cm}^2 \)
- Distance between the shorter sides: \( 11.98 \text{ cm} \)

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