To find the area of a right triangle, you can use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base is \(1 \frac{1}{2}\) centimeters, which can be converted to an improper fraction:
\[ 1 \frac{1}{2} = \frac{3}{2} \]
The height is \(2 \frac{1}{2}\) centimeters, which can also be converted to an improper fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \]
Now, substitute these values into the area formula:
\[ \text{Area} = \frac{1}{2} \times \frac{3}{2} \times \frac{5}{2} \]
Calculating this step by step:
\[ \text{Area} = \frac{1}{2} \times \frac{15}{4} = \frac{15}{8} \text{ square centimeters} \]
Now let's convert \(\frac{15}{8}\) to a mixed number:
\[ \frac{15}{8} = 1 \frac{7}{8} \]
Thus, the area of the triangle is:
\[ \boxed{1 \frac{7}{8}} \text{ square centimeters} \]