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The figure above shows a triangle ABC in which the measure of ∠ABC, b∘is greater than 90∘
 If AB = 3 units and BC = 4 units, which of the following can be the measure of side AC?
  1. 6
  2. 7
  3. 8
 
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I, II and III
  • e)
    None of the above
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The figure above shows a triangle ABC in which the measure of ∠A...
Given:
  • An obtuse triangle, 2 of whose sides are AB = 3 and BC = 4 units
To find:
  • The possible measure of the side AC that is opposite to the obtuse angle
Approach & Working:
  • To answer the question, we need to think of the constraints on AC. From the theory of triangles, we can figure that there are 2 constraints on AC:
    • Constraint 1: Since triangle ABC is obtuse-angled at B, AC2 > AB2 + BC2
    • Constraint 2: The sum of any 2 sides of a triangle must be greater than the 3rd side. So, AB + BC > AC
  • So, we will work out the range of possible values of AC by applying these 2 constraints, and then see, which of the 3 given values fit this range.
  • Applying Constraint 1
    • AC2 > 32 + 42
    • AC2 > 9 + 16
    • AC2 > 25
    • So, AC > 5
  • Applying Constraint 2
    • AB + BC > AC
    • So, 3 + 4 > AC
    • 7 > AC
    • Rewritten as: AC < 7
  • Getting to the answer
    • Thus, we see that 5 < AC < 7
    • The only value, out of the 3 options, that satisfies this range is Option I
 
Thus, the correct answer is Option A
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The figure above shows a triangle ABC in which the measure of ∠ABC, b∘is greater than 90∘If AB = 3 units and BC = 4 units, which of the following can be the measure of side AC? 6 7 8a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'A'. Can you explain this answer?
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