<p>In a triangle \(ABC\), coordinates of \(A\) are \((1, 2)\) and the equations of the medians through \(B\) and \(C\) are respectively \(x + y = 5\) and \(x = 4\). Then area of \(\triangle ABC\) (in sq. units) is</p>
Step-by-Step Solution
Key Concept: Use the property that medians intersect at the centroid G, which divides each median in ratio 2:1 from vertex. Find B and C by using that they lie on their respective medians and satisfying the centroid condition with known point A.
<p><strong>Step 1:</strong> Let B = (b₁, b₂) and C = (c₁, c₂). The centroid G divides the medians, and all three medians pass through G.</p><p><strong>Step 2:</strong> Median through B: x + y = 5, so B lies on this line: b₁ + b₂ = 5</p><p>Median through C: x = 4, so C lies on this line: c₁ = 4</p><p><strong>Step 3:</strong> The median from A passes through the midpoint D of BC. Midpoint D = ((b₁+c₁)/2, (b₂+c₂)/2) = ((b₁+4)/2, (b₂+c₂)/2)</p><p><strong>Step 4:</strong> Centroid G = ((1+b₁+4)/3, (2+b₂+c₂)/3) = ((5+b₁)/3, (2+b₂+c₂)/3)</p><p><strong>Step 5:</strong> Since G lies on median through B (x + y = 5):<br/>(5+b₁)/3 + (2+b₂+c₂)/3 = 5<br/>5 + b₁ + 2 + b₂ + c₂ = 15<br/>b₁ + b₂ + c₂ = 8</p><p><strong>Step 6:</strong> Since b₁ + b₂ = 5, we get c₂ = 3. So C = (4, 3)</p><p><strong>Step 7:</strong> Since G lies on median through C (x = 4):<br/>(5+b₁)/3 = 4<br/>5 + b₁ = 12<br/>b₁ = 7</p><p><strong>Step 8:</strong> From b₁ + b₂ = 5: b₂ = -2. So B = (7, -2)</p><p><strong>Step 9:</strong> Area of triangle ABC with A(1,2), B(7,-2), C(4,3):<br/>Area = ½|1(-2-3) + 7(3-2) + 4(2-(-2))|<br/>= ½|-5 + 7 + 16|<br/>= ½|18| = 9</p><p>∴ Answer: D (9 sq. units)</p>
Correct Answer: D