<p>If the area of the triangle with vertices <span class="math">(x, 0)</span>, <span class="math">(1, 1)</span> and <span class="math">(0, 2)</span> is 4 sq units, then the value of <span class="math">x</span> is</p>
Step-by-Step Solution
Key Concept: The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is calculated using the determinant formula: Area = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|. We use this formula with the given area of 4 sq units to find x.
<p><strong>Step 1:</strong> Apply the triangle area formula using vertices A(x, 0), B(1, 1), and C(0, 2).</p><p>Area = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|</p><p>4 = ½|x(1 - 2) + 1(2 - 0) + 0(0 - 1)|</p><p><strong>Step 2:</strong> Simplify the expression inside the absolute value.</p><p>4 = ½|x(-1) + 1(2) + 0|</p><p>4 = ½|-x + 2|</p><p><strong>Step 3:</strong> Multiply both sides by 2.</p><p>8 = |-x + 2|</p><p><strong>Step 4:</strong> Solve the absolute value equation by considering both cases.</p><p><strong>Case 1:</strong> -x + 2 = 8</p><p>-x = 6</p><p>x = -6</p><p><strong>Case 2:</strong> -x + 2 = -8</p><p>-x = -10</p><p>x = 10</p><p><strong>Step 5:</strong> Check which solution is among the given options.</p><p>From the options provided, x = -6 is option (c).</p><p><strong>∴ Answer:</strong> c</p>
Correct Answer: c