A trapezium PQRS inscribes a circle which touches the circle at M, A, N, B, Radius of circle is 10 cm. The length of each non-parallel side is 21 cm. What is the perimeter of trapezium?

Options

A. A)82 cm
B. B)84 cm
C. C)85.5 cm
D. D)can't be determined

Answer

To find the perimeter of trapezium PQRS that inscribes a circle, we can follow these steps:

### Step 1: Understand the properties of the trapezium
Since trapezium PQRS inscribes a circle, it means that the sum of the lengths of the opposite sides are equal. This can be expressed as:
\[ PQ + SR = PS + QR \]

### Step 2: Assign known values
From the problem, we know:
- The radius of the circle (r) = 10 cm
- The lengths of the non-parallel sides (PS and QR) = 21 cm each.

### Step 3: Calculate the sum of the lengths of the non-parallel sides
Since both non-parallel sides are equal:
\[ PS + QR = 21 + 21 = 42 \, \text{cm} \]

### Step 4: Use the property of the trapezium to find the lengths of the parallel sides
Let \( PQ = x \) and \( SR = y \). According to the property of the trapezium:
\[ x + y = 42 \]

### Step 5: Express the perimeter of the trapezium
The perimeter (P) of trapezium PQRS is the sum of all its sides:
\[ P = PQ + QR + RS + PS \]
Substituting the known values:
\[ P = x + 21 + y + 21 \]
\[ P = x + y + 42 \]

### Step 6: Substitute the value of \( x + y \)
From Step 4, we know that \( x + y = 42 \):
\[ P = 42 + 42 \]
\[ P = 84 \, \text{cm} \]

### Conclusion
The perimeter of trapezium PQRS is **84 cm**.

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