In ⧍ABC, AB = 2x + 5, BC = 3x–2, AC = 4x-8. Find all possible values of x. Then name the smallest and largest angles of ⧍XYZ.
Answers
Given : ⧍ABC, AB = 2x + 5, BC = 3x–2, AC = 4x-8
To Find : all possible values of x.
Solution:
AB = 2x + 5, BC = 3x–2, AC = 4x-8
Sum of any two sides > third side
=> AB + BC > AC
=> 2x + 5 + 3x - 2 > 4x - 8
=> x > - 11
AB + AC > BC
=> 2x + 5 + 4x - 8 > 3x - 2
=> 3x > 1
=> x > 1/3
AC + BC > AB
=> 4x - 8 + 3x - 2 > 2x + 5
=> 5x > 15
=> x > 3
x > - 11 , x > 1/3 , x > 3
Hence x > 3
All possible value of x ∈ (3 , ∞ )
smallest and largest angle will depend upon value of x
x = 4 => AB = 13 , BC = 10 , AC = 4
Largest angle = C and smallest angle = B
x = 6.5 => AB = 18 BC = 17.5 , AC= 18
Smallest angle = A and largest angle C & B are equal
x = 10 => AB = 25 , BC = 28 , AC = 32
=> Largest angle = B and smallest angle = C
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