Two adjacent sides of a parallelogram are 21 cms and 20 cms. The diagonal joining the end points of these two sides is 29 cms. The area of the parallelogram (in sq. cms) is

Options

A. 240
B. 120
C. 210
D. 420

Answer

To find the area of the parallelogram with adjacent sides of lengths 21 cm and 20 cm, and a diagonal of length 29 cm, we can follow these steps:

### Step 1: Understand the Geometry
We have a parallelogram where:
- One side (let's call it AB) = 21 cm
- The adjacent side (let's call it AD) = 20 cm
- The diagonal (let's call it AC) = 29 cm

### Step 2: Check if the Triangle is Right-Angled
We can check if triangle ABC (formed by sides AB, AD, and diagonal AC) is a right triangle by using the Pythagorean theorem:
\[ AC^2 = AB^2 + AD^2 \]
Substituting the values:
\[ 29^2 = 21^2 + 20^2 \]
Calculating:
\[ 841 = 441 + 400 \]
\[ 841 = 841 \]
Since the equation holds true, triangle ABC is a right triangle.

### Step 3: Calculate the Area of the Triangle
The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take AB as the base and AD as the height:
\[ \text{Area} = \frac{1}{2} \times 21 \times 20 \]
Calculating:
\[ \text{Area} = \frac{1}{2} \times 420 = 210 \text{ cm}^2 \]

### Step 4: Calculate the Area of the Parallelogram
Since the parallelogram consists of two identical triangles (ABC and ADC), the area of the parallelogram is:
\[ \text{Area of Parallelogram} = 2 \times \text{Area of Triangle ABC} \]
\[ \text{Area of Parallelogram} = 2 \times 210 = 420 \text{ cm}^2 \]

### Final Answer
The area of the parallelogram is **420 cm²**.

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