Problem: In triangle ABC, side a=3, side b=3, and side c>3. Let x be the largest number such that the magnitude, in degrees, of the angle opposite side c exceeds x. Then x equals:
Answer Choices:
A. 150
B. 120
C. 105
D. 90
E. 60
Solution:
Consider the triangle ABC where a=3,b=3, and c=3, and CD is the altitude to AB.∴BD=121 and AD=121. Then CD2=3−(23)2=43, and CD=23,∴∠B=∠A=30∘ and ∠C=120∘. Since side c is given greater than 3,∠C exceeds 120∘. (If two triangles have two sides of one equal to two sides of the other, and the third side of the first is greater than the third side of the second, the included angle of the first is greater than the included angle of the second.) c2=a2+b2−2abcosC,cosC=2aba2+b2−c2=2333+3−c2 where c2>9.∴cosC<−21, so that ∠C>120∘.
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