<p>The base of an isosceles triangle is the intercept made by the line \(x + 2y = 4\) with the coordinate axes. If the equations of the equal sides be \(x = 4\) and \(y = mx + c\) then find the value of \(8m + c\).</p>
Step-by-Step Solution
Key Concept: Find the base vertices by determining where the line x + 2y = 4 intersects the axes, then use the property that in an isosceles triangle, the two equal sides must be equidistant from the base vertices. The vertex where the two equal sides meet must satisfy symmetry conditions.
<p><strong>Step 1:</strong> Find the intercepts of x + 2y = 4 with the coordinate axes.</p><p>When y = 0: x = 4, giving point A(4, 0)</p><p>When x = 0: 2y = 4, so y = 2, giving point B(0, 2)</p><p>The base AB has endpoints A(4, 0) and B(0, 2).</p><p><strong>Step 2:</strong> Identify the equal sides. We're given that the two equal sides are x = 4 and y = mx + c.</p><p>The line x = 4 is vertical and passes through A(4, 0).</p><p>Therefore, one equal side is along x = 4 starting from A(4, 0).</p><p><strong>Step 3:</strong> The other equal side y = mx + c must pass through B(0, 2).</p><p>Substituting: 2 = m(0) + c, so c = 2.</p><p><strong>Step 4:</strong> For an isosceles triangle, both equal sides must meet at apex vertex P. The apex lies on x = 4, so P has coordinates (4, y_p) for some y_p.</p><p>Since P also lies on y = mx + c: y_p = m(4) + 2, giving y_p = 4m + 2.</p><p><strong>Step 5:</strong> Use the isosceles condition: |PA| = |PB|.</p><p>|PA| = |y_p - 0| = |4m + 2|</p><p>|PB| = √[(4-0)² + (4m+2-2)²] = √[16 + (4m)²] = √[16 + 16m²]</p><p><strong>Step 6:</strong> Set |PA|² = |PB|²:</p><p>(4m + 2)² = 16 + 16m²</p><p>16m² + 16m + 4 = 16 + 16m²</p><p>16m + 4 = 16</p><p>16m = 12</p><p>m = 3/4</p><p><strong>Step 7:</strong> Calculate 8m + c.</p><p>8m + c = 8(3/4) + 2 = 6 + 2 = 8</p><p><strong>∴ Answer: 8m + c = 8</strong></p>
Correct Answer: 8