<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 from line-axes intercepts, then use the isosceles triangle property that the two equal sides are equidistant from the apex. The apex lies on x = 4, and symmetry of equal sides determines the relationship between m and c.
<p><strong>Step 1: Find the base vertices</strong></p><p>The line x + 2y = 4 intersects the coordinate axes at:</p><p>• x-axis (y = 0): x = 4, giving point A(4, 0)</p><p>• y-axis (x = 0): 2y = 4, so y = 2, giving point B(0, 2)</p><p>The base AB connects (4, 0) and (0, 2).</p><p><strong>Step 2: Find the midpoint of base AB</strong></p><p>Midpoint M = ((4+0)/2, (0+2)/2) = (2, 1)</p><p><strong>Step 3: Use the isosceles property</strong></p><p>One equal side is x = 4 (vertical line). The apex C lies on x = 4, so C = (4, k) for some k.</p><p>For an isosceles triangle with CA = CB, both equal sides must make equal angles with the base or be symmetric about the perpendicular bisector of AB.</p><p><strong>Step 4: Find slope of base AB</strong></p><p>Slope of AB = (2 - 0)/(0 - 4) = -1/2</p><p>Slope of perpendicular bisector of AB = 2 (negative reciprocal)</p><p><strong>Step 5: Determine the apex C</strong></p><p>The perpendicular bisector passes through M(2, 1) with slope 2:</p><p>y - 1 = 2(x - 2) → y = 2x - 3</p><p>Since C lies on x = 4: y = 2(4) - 3 = 5</p><p>So C = (4, 5)</p><p><strong>Step 6: Verify with the second equal side</strong></p><p>The second equal side y = mx + c passes through both C(4, 5) and B(0, 2).</p><p>Slope m = (5 - 2)/(4 - 0) = 3/4</p><p>Using point B(0, 2): 2 = m(0) + c → c = 2</p><p><strong>Step 7: Verify distances CA and CB</strong></p><p>CA: from C(4, 5) to A(4, 0) = 5 ✓</p><p>CB: from C(4, 5) to B(0, 2) = √[(4-0)² + (5-2)²] = √(16 + 9) = √25 = 5 ✓</p><p><strong>Step 8: Calculate 8m + c</strong></p><p>8m + c = 8(3/4) + 2 = 6 + 2 = 8</p><p><strong>∴ Answer: 8</strong></p>
Correct Answer: 8