Straight Lines
Coordinate Geometry
Grade 11
Question:
<p>Find the coordinates of point <i>F</i> given that <i>\frac{BF}{CF} = \frac{BE}{AC} = \frac{a}{4}</i> where <i>B(0,a)</i>, <i>C(a,a)</i>, and <i>A(a,0)</i>.</p>
<p>(A) <i>\left(-\frac{a}{3}, a\right)</i></p>
<p>(B) <i>\left(\frac{a}{3}, a\right)</i></p>
<p>(C) <i>\left(0, \frac{a}{3}\right)</i></p>
<p>(D) <i>\left(\frac{a}{4}, a\right)</i></p>
Step-by-Step Solution
Key Concept: Use the given ratio to determine the horizontal displacement of F from B, maintaining the same y-coordinate.
<p><strong>Step 1:</strong> From the similarity condition and the ratio <i>\frac{BF}{CF} = \frac{a}{4}</i>, we have:</p><p>\[BF = \frac{a}{3}\]</p><p><strong>Step 2:</strong> Since <i>F</i> lies on the vertical line through <i>B(0,a)</i> (at the same height), and <i>BF = \frac{a}{3}</i> measured horizontally to the left:</p><p>\[F = \left(-\frac{a}{3}, a\right)\]</p><p><strong>Step 3:</strong> With <i>a = 7</i>:</p><p>\[F = \left(-\frac{7}{3}, 7\right)\]</p><p>∴ Answer is <i>(A) \left(-\frac{a}{3}, a\right)</i>.</p>
Correct Answer: A