The base of a right prism is a quadrilateral ABCD. Given that AB = 9 cm, BC = 14 cm, CD = 13 cm, DA = 12 cm and `angleDAB = 90^(@)`. If the volume of the prism be `2070 cm^(3)`, then the area of the lateral surface is

Options

A. `720 cm^(2)`
B. `810 cm^(2)`
C. `1260 cm^(2)`
D. `2070 cm^(2)`

Answer

To solve the problem of finding the lateral surface area of a right prism with a quadrilateral base ABCD, we will follow these steps:

### Step 1: Identify the dimensions of the quadrilateral
Given:
- AB = 9 cm
- BC = 14 cm
- CD = 13 cm
- DA = 12 cm
- Angle DAB = 90°

### Step 2: Divide the quadrilateral into two triangles
We can divide quadrilateral ABCD into two triangles: Triangle ABD and Triangle BCD.

### Step 3: Calculate the area of Triangle ABD
Triangle ABD is a right triangle (since angle DAB = 90°). We can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, AB is the base and AD is the height.
\[ \text{Area of } \triangle ABD = \frac{1}{2} \times AB \times AD = \frac{1}{2} \times 9 \times 12 = 54 \, \text{cm}^2 \]

### Step 4: Calculate the length of diagonal BD using Pythagorean theorem
Using the Pythagorean theorem in triangle ABD:
\[ BD = \sqrt{AB^2 + AD^2} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \, \text{cm} \]

### Step 5: Calculate the area of Triangle BCD using Heron's formula
First, calculate the semi-perimeter (s) of triangle BCD:
\[ s = \frac{BC + CD + BD}{2} = \frac{14 + 13 + 15}{2} = 21 \, \text{cm} \]

Now, apply Heron's formula:
\[ \text{Area} = \sqrt{s(s - BC)(s - CD)(s - BD)} \]
\[ \text{Area of } \triangle BCD = \sqrt{21(21 - 14)(21 - 13)(21 - 15)} \]
\[ = \sqrt{21 \times 7 \times 8 \times 6} \]
Calculating this gives:
\[ = \sqrt{21 \times 336} = \sqrt{7056} = 84 \, \text{cm}^2 \]

### Step 6: Calculate the total area of the base (quadrilateral ABCD)
\[ \text{Area of ABCD} = \text{Area of } \triangle ABD + \text{Area of } \triangle BCD \]
\[ = 54 + 84 = 138 \, \text{cm}^2 \]

### Step 7: Find the height of the prism
Using the formula for volume:
\[ \text{Volume} = \text{Area of base} \times \text{Height} \]
Given that the volume is 2070 cm³:
\[ 2070 = 138 \times \text{Height} \]
\[ \text{Height} = \frac{2070}{138} = 15 \, \text{cm} \]

### Step 8: Calculate the perimeter of the base
\[ \text{Perimeter} = AB + BC + CD + DA = 9 + 14 + 13 + 12 = 48 \, \text{cm} \]

### Step 9: Calculate the lateral surface area
The lateral surface area (LSA) of the prism is given by:
\[ \text{LSA} = \text{Perimeter} \times \text{Height} \]
\[ \text{LSA} = 48 \times 15 = 720 \, \text{cm}^2 \]

### Final Answer
The area of the lateral surface is **720 cm²**.

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