PQRS is an isosceles trapezium in which PS = 10 cm, PQ = SR = 13 cm and the distance between PS and QR is 12 cm. Find the area of the trapezium.

Options

A. 180 `cm^(2)`
B. 160 `cm^(2)`
C. 176 `cm^(2)`
D. 194 `cm^(2)`

Answer

To find the area of the isosceles trapezium PQRS, we will follow these steps:

### Step 1: Identify the given dimensions
We have:
- PS = 10 cm (one of the parallel sides)
- PQ = SR = 13 cm (the non-parallel sides)
- The distance between the parallel sides PS and QR (height, h) = 12 cm

### Step 2: Use the formula for the area of a trapezium
The area \( A \) of a trapezium can be calculated using the formula:
\[ A = \frac{1}{2} \times (a + b) \times h \]
where \( a \) and \( b \) are the lengths of the two parallel sides, and \( h \) is the height.

### Step 3: Find the length of the other parallel side QR
Since PQRS is an isosceles trapezium, we can drop perpendiculars from points Q and R to line PS. Let the foot of the perpendiculars be points M and N respectively. We have:
- PM = NR = x (the segments on PS)
- PS = 10 cm, so \( PM + MN + NR = 10 \)
- MN = QR (the length we need to find)

Using the Pythagorean theorem in triangle SMR:
\[ PQ^2 = h^2 + MR^2 \]
Substituting the known values:
\[ 13^2 = 12^2 + MR^2 \]
\[ 169 = 144 + MR^2 \]
\[ MR^2 = 169 - 144 = 25 \]
\[ MR = 5 \text{ cm} \]

### Step 4: Calculate the length of QR
Since MR = 5 cm and is equal to NM (by symmetry in the isosceles trapezium), we have:
\[ MN = PS - PM - NR = 10 - 5 - 5 = 0 \]
Thus, QR = 10 cm.

### Step 5: Calculate the area of trapezium PQRS
Now we can substitute the values into the area formula:
\[ A = \frac{1}{2} \times (PS + QR) \times h \]
\[ A = \frac{1}{2} \times (10 + 10) \times 12 \]
\[ A = \frac{1}{2} \times 20 \times 12 \]
\[ A = 10 \times 12 = 120 \text{ cm}^2 \]

### Final Answer
The area of trapezium PQRS is \( 120 \text{ cm}^2 \).