Trigonometry & Inverse Trigonometry
Pedal Triangle
Grade 11
Question:
<p>If in the pedal triangle DEF of an acute-angled triangle ABC, the sides are denoted by <i>l</i>, <i>m</i>, <i>n</i>, then <i>l</i>/(<i>a</i>²) + <i>m</i>/(<i>b</i>²) + <i>n</i>/(<i>c</i>²) is equal to</p>
<p>(a) <i>(a</i>² + <i>b</i>² + <i>c</i>²)/(abc)</p>
<p>(b) <i>(a</i>² + <i>b</i>² + <i>c</i>²)/(2abc)</p>
<p>(c) [Other option]</p>
<p>(d) [Other option]</p>
Step-by-Step Solution
Key Concept: The pedal triangle has sides equal to a cos A, b cos B, c cos C; apply this formula and use properties of triangle geometry.
<p><strong>Step 1:</strong> In an acute-angled triangle ABC with pedal triangle DEF, the sides of the pedal triangle are: <i>l</i> = <i>a</i>cos A, <i>m</i> = <i>b</i>cos B, <i>n</i> = <i>c</i>cos C</p><p><strong>Step 2:</strong> Computing the sum:</p><p><i>l</i>/(<i>a</i>²) + <i>m</i>/(<i>b</i>²) + <i>n</i>/(<i>c</i>²) = (a cos A)/(<i>a</i>²) + (b cos B)/(<i>b</i>²) + (c cos C)/(<i>c</i>²)</p><p><strong>Step 3:</strong> Simplifying: = (cos A)/<i>a</i> + (cos B)/<i>b</i> + (cos C)/<i>c</i></p><p><strong>Step 4:</strong> Using the property in triangles and extended sine rule, this equals <i>(a</i>² + <i>b</i>² + <i>c</i>²)/(2abc)</p><p>∴ Answer is (b).</p>
Correct Answer: a