1. **Stating the problem:** We have a right triangle with legs of lengths 12 cm and 44 cm, and the hypotenuse labeled as $5y$ cm. We need to find the value of $a$ (assuming $a$ is the length of the hypotenuse or related to it). 2. **Formula used:** In a right triangle, the Pythagorean theorem applies: $$a^2 = b^2 + c^2$$ where $a$ is the hypotenuse, and $b$, $c$ are the legs. 3. **Applying the theorem:** Here, the hypotenuse is $5y$, and the legs are 12 and 44: $$ (5y)^2 = 12^2 + 44^2 $$ 4. **Calculate the squares:** $$ 25y^2 = 144 + 1936 $$ $$ 25y^2 = 2080 $$ 5. **Solve for $y^2$:** $$ y^2 = \frac{2080}{25} $$ 6. **Simplify the fraction:** $$ y^2 = \frac{\cancel{2080}}{\cancel{25}} = 83.2 $$ 7. **Find $y$ by taking the square root:** $$ y = \sqrt{83.2} \approx 9.12 $$ 8. **Find $a$ if $a = 5y$:** $$ a = 5 \times 9.12 = 45.6 $$ **Final answer:** $$ a = 45.6 \text{ cm} $$