Trigonometry & Inverse Trigonometry
Triangle Geometry
Grade 11
Question:
<p>Let <i>A</i>(7,0), <i>B</i>(4,4) and <i>C</i>(0,0) and triangle <i>DEF</i> is isosceles with <i>DE</i> = <i>DF</i>. Then, the curve on which <i>F</i> may lie</p>
<p>(a) <i>x</i> = 4 or <i>x</i> + <i>y</i> = 7 or 4<i>x</i> = 3<i>y</i></p>
<p>(b) <i>x</i> = 4 or \(x^2 + y^2 = 4x + 4y\)</p>
<p>(c) \(3(x^2 + y^2) + 196 = 49(x + y)\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: The locus of foot of perpendicular satisfying isosceles constraints is determined by perpendicularity conditions and geometric symmetry.
<p>Using the condition <i>DE</i> = <i>DF</i> and the constraint that <i>F</i> is the foot of perpendicular from <i>P</i> onto side <i>AB</i>, the locus of <i>F</i> must satisfy one of three conditions: <i>x</i> = 4 (perpendicular to horizontal through vertex), <i>x</i> + <i>y</i> = 7 (perpendicular to line <i>AB</i>), or 4<i>x</i> = 3<i>y</i> (angle bisector condition).</p>
Correct Answer: A