Question:

<p><strong>Step 1:</strong> Set up notation. Let BC = a, CA = b, and AB = c.</p><p><strong>Step 2:</strong> Write the given constraints:<br/>• Constraint 1: b + a = 20 (given: BC + CA = 20)<br/>• Constraint 2: c + a = 21 (given: AB + BC = 21)<br/>• Constraint 3: Apply Law of Cosines with ∠A = 120°<br/>a² = b² + c² - 2bc·cos(120°)<br/>a² = b² + c² - 2bc·(-1/2)<br/>a² = b² + c² + bc</p><p><strong>Step 3:</strong> From Constraint 1: b = 20 - a<br/>From Constraint 2: c = 21 - a</p><p><strong>Step 4:</strong> Substitute into the Law of Cosines equation:<br/>a² = (20 - a)² + (21 - a)² + (20 - a)(21 - a)</p><p><strong>Step 5:</strong> Expand (20 - a)²:<br/>(20 - a)² = 400 - 40a + a²</p><p><strong>Step 6:</strong> Expand (21 - a)²:<br/>(21 - a)² = 441 - 42a + a²</p><p><strong>Step 7:</strong> Expand (20 - a)(21 - a):<br/>(20 - a)(21 - a) = 420 - 20a - 21a + a² = 420 - 41a + a²</p><p><strong>Step 8:</strong> Substitute all expansions:<br/>a² = (400 - 40a + a²) + (441 - 42a + a²) + (420 - 41a + a²)<br/>a² = 400 + 441 + 420 - 40a - 42a - 41a + a² + a² + a²<br/>a² = 1261 - 123a + 3a²</p><p><strong>Step 9:</strong> Rearrange:<br/>a² - 3a² + 123a - 1261 = 0<br/>-2a² + 123a - 1261 = 0<br/>2a² - 123a + 1261 = 0</p><p><strong>Step 10:</strong> Use the quadratic formula:<br/>a = [123 ± √(123² - 4(2)(1261))] / (2·2)<br/>a = [123 ± √(15129 - 10088)] / 4<br/>a = [123 ± √5041] / 4<br/>a = [123 ± 71] / 4</p><p><strong>Step 11:</strong> Calculate both solutions:<br/>a = (123 + 71)/4 = 194/4 = 48.5<br/>a = (123 - 71)/4 = 52/4 = 13</p><p><strong>Step 12:</strong> Check validity. If a = 48.5, then b = 20 - 48.5 = -28.5 (negative, invalid). If a = 13, then b = 20 - 13 = 7 and c = 21 - 13 = 8 (all positive, valid).</p><p><strong>Step 13:</strong> Verify: a² = b² + c² + bc gives 169 = 49 + 64 + 56 = 169 ✓</p><p><strong>∴ Answer:</strong> A</p>