The base and altitude of a right-angled triangle are $12\text{ cm}$ and $5\text{ cm}$ respectively. The perpendicular distance of its hypotenuse from the opposite vertex is

### Concept & Properties of Right Triangles The "perpendicular distance of the hypotenuse from the opposite vertex" is precisely the altitude drawn to the hypotenuse. For a right-angled triangle with perpendicular legs $p$ and $b$, and hypotenuse $h$, the length of this altitude is given by: $$ \text{Altitude} = \frac{p \times b}{h} $$ ### Step-by-Step Solution 1. Identify the given base ($b = 12\text{ cm}$) and altitude/perpendicular ($p = 5\text{ cm}$). 2. Calculate the hypotenuse ($h$) using the Pythagorean theorem: $h = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\text{ cm}$. 3. Calculate the perpendicular distance to the hypotenuse: $\text{Distance} = \frac{12 \times 5}{13}$ $\text{Distance} = \frac{60}{13}\text{ cm}$. 4. Convert the improper fraction to a mixed number: $\frac{60}{13} = 4 \text{ with a remainder of } 8$, which is $4\frac{8}{13}\text{ cm}$. ### Exam Strategy & Shortcut Recognize 5-12-13 as a standard Pythagorean triplet immediately. The product of the legs divided by the hypotenuse yields $\frac{60}{13}$. The nearest multiple of 13 is $13 \times 4 = 52$, leaving a remainder of 8, pointing straight to $4\frac{8}{13}$. ### Common Pitfall A common trap is mistaking the "altitude" given in the problem statement ($5\text{ cm}$) for the altitude to the hypotenuse. Read carefully: it refers to the legs of the right triangle. ### Final Answer Therefore, the correct answer is **$4\frac{8}{13}\text{ cm}$**.