Trigonometry & Inverse Trigonometry
Area and Inradius of Triangle
Grade 11
Question:
<p><strong>Ex. 23:</strong> <strong>Statement I</strong> If the sides of a triangle are 13, 14, 15 then the radius of incircle = 4</p><p><strong>Statement II</strong> In triangle ABC, \(A = \sqrt{s(s-a)(s-b)(s-c)}\) where \(s = \dfrac{a+b+c}{2}\) and \(r = \dfrac{A}{s}\)</p>
<p>(a) Statement I is True, Statement II is True; Statement II is a correct explanation for Statement I.</p>
<p>(b) Statement I is True, Statement II is True; Statement II is NOT a correct explanation for Statement I.</p>
<p>(c) Statement I is True, Statement II is False.</p>
<p>(d) Statement I is False, Statement II is True.</p>
Step-by-Step Solution
Key Concept: The inradius formula \(r = \frac{A}{s}\) combined with Heron's formula \(A = \sqrt{s(s-a)(s-b)(s-c)}\) allows direct calculation of the inradius from the three sides.
<p><strong>Step 1:</strong> For a triangle with sides \(a = 13, b = 14, c = 15\):</p><p><strong>Step 2:</strong> \(s = \dfrac{13+14+15}{2} = 21\)</p><p><strong>Step 3:</strong> \(A = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84\)</p><p><strong>Step 4:</strong> \(r = \dfrac{A}{s} = \dfrac{84}{21} = 4\)</p><p>∴ Answer is (a).</p>
Correct Answer: a