The area of a right-angled triangle is $40$ times its base. What is its height?

### Concept & Algebraic Formulation Translate the textual conditions directly into an algebraic equation using the standard area formula. A right-angled triangle uses its perpendicular legs as the base and height. $$ \text{Area} = \frac{1}{2} \times b \times h $$ ### Step-by-Step Solution * Let the base of the triangle be $b$ and its height be $h$. * According to the given condition, Area = $40 \times b$. * We also know the standard formula: Area = $\frac{1}{2} \times b \times h$. * Equate the two expressions for area: $\frac{1}{2} \times b \times h = 40 \times b$. * Assuming the base $b$ is non-zero, divide both sides by $b$: $\frac{1}{2} \times h = 40$. * Multiply both sides by 2 to isolate $h$: $h = 80$. ### Exam Strategy & Shortcut Recognize that if Area = $k \times \text{base}$, then $\frac{1}{2} \times \text{height} = k$. Therefore, the height is simply always $2k$. Here, $k = 40$, so height is $2 \times 40 = 80$. This can be solved mentally in two seconds. ### Common Pitfall Choosing "Data inadequate" because only one numerical value ($40$) is given. Realizing the variable for base ($b$) cancels out is the key insight to avoiding this trap. ### Final Answer Therefore, the correct answer is **$80\text{ cm}$**.