<p>Two poles of heights 20 m and 80 m are standing on a horizontal ground. The height (in metres) of the point of intersection of the lines joining the top of each pole to the foot of the other pole is:</p>
Step-by-Step Solution
Key Concept: Use similar triangles formed by the intersecting lines. The point of intersection divides the vertical plane into similar triangular regions whose heights can be related through the ratio of pole heights.
<p><strong>Step 1:</strong> Set up coordinates. Place pole 1 (height 20 m) at origin and pole 2 (height 80 m) at distance d on the ground.</p><p><strong>Step 2:</strong> Line from top of pole 1 to foot of pole 2: passes through (0, 20) and (d, 0). Equation: y = 20 - (20/d)x</p><p><strong>Step 3:</strong> Line from top of pole 2 to foot of pole 1: passes through (d, 80) and (0, 0). Equation: y = (80/d)x</p><p><strong>Step 4:</strong> At intersection point, both equations are equal: 20 - (20/d)x = (80/d)x</p><p><strong>Step 5:</strong> Solving: 20 = (100/d)x → x = d/5</p><p><strong>Step 6:</strong> Substitute back into y = (80/d)x: y = (80/d)(d/5) = 80/5 = 16</p><p><strong>Alternative Formula:</strong> For two poles of heights h₁ and h₂, intersection height = h₁h₂/(h₁ + h₂) = (20 × 80)/(20 + 80) = 1600/100 = 16</p><p>∴ Answer: <strong>16 metres</strong></p>
Correct Answer: 16