Directions : Each of the questions below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. What is the area of the isosceles triangle ? I. Perimeter of the isosceles triangle is $18$ metres. II. Base of the triangle is $8$ metres. III. Height of the triangle is $3$ metres.

### Concept & Isosceles Triangle Properties To find the area of an isosceles triangle, we typically need its base ($b$) and height ($h$): $$ \text{Area} = \frac{1}{2} \times b \times h $$ The equal sides ($a$), base, and height are related by the Pythagorean theorem because the altitude drawn to the base bisects it, creating two right-angled triangles: $$ a^2 = h^2 + \left(\frac{b}{2}\right)^2 $$ The perimeter is given by $2a + b$. ### Step-by-Step Solution Let's test combinations of two statements to see if they provide enough info to find the area: - **Statements II & III:** We are given Base $= 8$ and Height $= 3$. Area $= \frac{1}{2} \times 8 \times 3 = 12$. Sufficient. - **Statements I & II:** We are given Perimeter $= 18$ and Base $= 8$. $$ 2a + 8 = 18 \implies 2a = 10 \implies a = 5 $$ With side $5$ and base $8$, we find height $h = \sqrt{5^2 - 4^2} = 3$. Area can now be calculated. Sufficient. - **Statements I & III:** We are given Perimeter $= 18$ and Height $= 3$. We know $2a + b = 18 \implies a = 9 - \frac{b}{2}$. Squaring both sides gives: $a^2 = 81 - 9b + \frac{b^2}{4}$. Substituting this into our Pythagorean equation ($a^2 = 3^2 + \frac{b^2}{4}$): $$ 81 - 9b + \frac{b^2}{4} = 9 + \frac{b^2}{4} $$ $$ 81 - 9b = 9 \implies 9b = 72 \implies b = 8 $$ With base $8$ and height $3$, the area can be calculated. Sufficient. Since any pair of statements provides enough information to calculate the base and height, any two statements are sufficient. ### Exam Strategy & Shortcut An isosceles triangle is mathematically defined by exactly two independent parameters. The problem gives three distinct, valid metrics (perimeter, base, height). Combining any two of these will logically lock the triangle into a single specific size and shape, allowing you to find the area. ### Common Pitfall Assuming that statements I and III combined require complex algebra that might yield multiple solutions (a quadratic equation). As shown in the derivation, the $b^2$ terms actually cancel out cleanly, leaving a single linear solution. ### Final Answer Therefore, the correct answer is **Any two of the three**.