Directions: Each of the questions given below is followed by three statements. You have to study the question and all the three statements given to decide whether any information provided in the statement(s) is/are redundant and can be dispensed with while answering the given question. What is the height of the triangle whose area is equal to the area of a rectangle? I. The ratio between the length and breadth of the rectangle is $3 : 2$. II. The base of the triangle is $16$ cm. III. The perimeter of the rectangle is $80$ cm.

### Concept & Equating Areas To find the unknown height of the triangle, we need its base and the exact area of the rectangle. Area of Triangle = Area of Rectangle $$\frac{1}{2} \times b \times h = l \times w$$ ### Step-by-Step Solution 1. **From Statement II:** The base of the triangle is $b = 16$. Substituting this into our main equation gives $\frac{1}{2} \times 16 \times h = l \times w \implies 8h = l \times w$. We still need to find the numerical area of the rectangle ($l \times w$). 2. **From Statement I:** The ratio of length to breadth is $3 : 2$. Let length $l = 3x$ and breadth $w = 2x$. The area is $6x^2$. We need the specific value of the multiplier $x$. 3. **From Statement III:** The perimeter of the rectangle is $80$ cm. Using the formula $2(l + w) = 80 \implies 2(3x + 2x) = 80 \implies 10x = 80 \implies x = 8$. 4. Now we can resolve the rectangle's dimensions: $l = 24$ and $w = 16$. The area of the rectangle is $24 \times 16 = 384$. 5. Returning to our first equation: $8h = 384 \implies h = 48$ cm. 6. Since every single statement provided an indispensable piece of the puzzle (triangle base, rectangle ratio, rectangle perimeter), no statement can be skipped. ### Exam Strategy & Shortcut Count your variables and your independent clues. You need to find the height, which depends on the triangle base ($1$ variable) and the rectangle dimensions ($2$ variables). We have three unknowns in total. Statement II gives the base. Statements I and III give two independent geometric properties of the rectangle (ratio and perimeter). Three unknowns with three independent constraints means all clues are strictly required. ### Common Pitfall Assuming that because the question asks "which is redundant", there must automatically be a redundant statement. Always trace the dependency of variables to ensure no single statement can actually be bypassed. ### Final Answer Therefore, the correct answer is **All I, II. and III together are required**.