PQRS is a parallelogram with `PQ = 15 cm, ST = 10 cm, and QV = 12 cm.` Find PS.

Answer

To find the length of PS in the parallelogram PQRS, we can use the properties of the area of a parallelogram. Here’s a step-by-step solution:

### Step 1: Understand the given information
We know:
- PQ = 15 cm (one side of the parallelogram)
- ST = 10 cm (the height corresponding to base PQ)
- QV = 12 cm (the height corresponding to base PS)

### Step 2: Calculate the area using base PQ and height ST
The area of a parallelogram can be calculated using the formula:
\[ \text{Area} = \text{Base} \times \text{Height} \]

Using PQ as the base:
\[ \text{Area} = PQ \times ST = 15 \, \text{cm} \times 10 \, \text{cm} = 150 \, \text{cm}^2 \]

### Step 3: Set up the area equation using base PS and height QV
Now, we can also express the area using PS as the base and QV as the height:
\[ \text{Area} = PS \times QV \]

Substituting the value of QV:
\[ \text{Area} = PS \times 12 \, \text{cm} \]

### Step 4: Equate the two area expressions
Since both expressions represent the area of the same parallelogram, we can set them equal to each other:
\[ 150 \, \text{cm}^2 = PS \times 12 \, \text{cm} \]

### Step 5: Solve for PS
To find PS, we rearrange the equation:
\[ PS = \frac{150 \, \text{cm}^2}{12 \, \text{cm}} \]

Calculating this gives:
\[ PS = 12.5 \, \text{cm} \]

### Final Answer
Thus, the length of PS is \( 12.5 \, \text{cm} \).

---