PQRS is a parallelogram. RA and RB are perpendiculars from R on PQ and PS, respectively. If RA = 15 cm, RB = 22 cm, and QR = 26 cm, find PQ.

Answer

To find the length of PQ in the parallelogram PQRS, we can use the area of the parallelogram calculated using two different pairs of base and height. Here’s the step-by-step solution:

### Step 1: Understand the problem
We have a parallelogram PQRS with perpendiculars RA and RB from point R to sides PQ and PS, respectively. We know the lengths of these perpendiculars and one side of the parallelogram.

### Step 2: Write down the known values
- RA (height from R to PQ) = 15 cm
- RB (height from R to PS) = 22 cm
- QR (length of side PS) = 26 cm
- We need to find PQ.

### Step 3: Use the area formula for the parallelogram
The area of a parallelogram can be calculated using two different pairs of base and height:
1. Area using base PQ and height RA:
\[ \text{Area} = PQ \times RA \]
2. Area using base PS and height RB:
\[ \text{Area} = PS \times RB \]

Since both expressions represent the same area, we can set them equal to each other:
\[ PQ \times RA = PS \times RB \]

### Step 4: Substitute the known values into the equation
Substituting the known values into the equation:
\[ PQ \times 15 = 26 \times 22 \]

### Step 5: Calculate the right side of the equation
Calculate \(26 \times 22\):
\[ 26 \times 22 = 572 \]
So, we have:
\[ PQ \times 15 = 572 \]

### Step 6: Solve for PQ
To find PQ, divide both sides by 15:
\[ PQ = \frac{572}{15} \]

### Step 7: Perform the division
Calculating \(572 \div 15\):
\[ PQ = 38.13 \text{ cm (approximately)} \]

### Final Answer:
The length of PQ is approximately **38.13 cm**.

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