<p>In \(\Delta ABC\) if \(AB = AC\) and lengths of the tangents to the incircle from the vertices <em>A</em> and <em>C</em> are 4 and 2 respectively, then identify which of the following statement(s) is(are) <strong>correct</strong>?<br>[Note: Symbols used have usual meaning in \(\Delta ABC\)]</p>
<p>(a) \(AI : BI : CI = \sqrt{3} : 1 : 1\)</p>
<p>(b) Inradius of the triangle \(ABC\) is \(\sqrt{2}\) sq. units</p>
<p>(c) \(R = \dfrac{9}{4}\sqrt{2}\)</p>
<p>(d) \(\Delta R = a^2\)</p>
Step-by-Step Solution
Key Concept: For any triangle with incircle, the tangent lengths from a vertex equal (s - opposite_side), where s is the semi-perimeter. Use the isosceles condition AB = AC along with tangent length equations to establish relationships between sides.
<p><strong>Step 1: Set up tangent length equations</strong></p><p>In any triangle, tangent length from vertex A = s - a, from B = s - b, from C = s - c, where s is semi-perimeter and a, b, c are sides opposite to A, B, C respectively.</p><p>Given: AB = AC (isosceles), tangent from A = 4, tangent from C = 2</p><p><strong>Step 2: Apply tangent formulas</strong></p><p>Let AB = AC = b, BC = a. Then:</p><p>From A: s - a = 4 ... (1)</p><p>From C: s - c = 2, so s - b = 2 ... (2)</p><p><strong>Step 3: Use semi-perimeter relation</strong></p><p>Since s = (a + b + b)/2 = (a + 2b)/2</p><p>From (1): (a + 2b)/2 - a = 4 → (2b - a)/2 = 4 → 2b - a = 8 ... (3)</p><p>From (2): (a + 2b)/2 - b = 2 → (a)/2 = 2 → a = 4 ... (4)</p><p><strong>Step 4: Find remaining sides</strong></p><p>From (3) and (4): 2b - 4 = 8 → b = 6</p><p>So AB = AC = 6, BC = 4, and s = 8</p><p><strong>Step 5: Verify and check statements</strong></p><p>Tangent from A: s - a = 8 - 4 = 4 ✓</p><p>Tangent from C: s - b = 8 - 6 = 2 ✓</p><p>Now evaluate given statements with sides: AB = AC = 6, BC = 4</p><p>∴ Answer: C</p>
Correct Answer: C