Straight Lines
Rotation of lines and triangle properties
Grade 11
Question:
<p>If the straight line \(3x - 4y + 7 = 0\) rotated through 90° about a point (3, 4) meets the coordinate axes at <em>A</em> and <em>B</em> and a \(\Delta AOB\) is formed (<em>O</em> is the origin), then:</p>
<p>(a) Area of the triangle formed by orthocentre, circumcentre and centroid of the \(\Delta AOB\) is 1 sq. units.</p>
<p>(b) Area of the triangle formed by orthocentre, circumcentre and incentre of the \(\Delta AOB\) is 1 sq. units.</p>
<p>(c) Distance between orthocentre and incentre is 2 units.</p>
<p>(d) Distance between orthocentre and circumcentre is 5 units.</p>
Step-by-Step Solution
Key Concept: When a line rotates 90° about a point, its slope becomes the negative reciprocal of the original slope. The rotated line passes through (3,4) with this new slope, then find its intercepts to determine the triangle area.
<p><strong>Step 1:</strong> Find the slope of the original line 3x - 4y + 7 = 0.</p><p>Rewriting: y = (3/4)x + 7/4, so m₁ = 3/4</p><p><strong>Step 2:</strong> After 90° rotation, the new slope is m₂ = -1/m₁ = -4/3</p><p><strong>Step 3:</strong> The rotated line passes through (3, 4) with slope -4/3:</p><p>y - 4 = -4/3(x - 3)</p><p>y - 4 = -4x/3 + 4</p><p>y = -4x/3 + 8</p><p>or 4x + 3y - 24 = 0</p><p><strong>Step 4:</strong> Find intercepts:</p><p>• x-intercept (y = 0): 4x = 24 ⟹ x = 6, so A = (6, 0)</p><p>• y-intercept (x = 0): 3y = 24 ⟹ y = 8, so B = (0, 8)</p><p><strong>Step 5:</strong> Area of △AOB with O at origin:</p><p>Area = (1/2) × |OA| × |OB| = (1/2) × 6 × 8 = 24 square units</p><p>∴ Answer: A</p>
Correct Answer: A