If a, b, c denote the sides of a `DeltaABC` such that
`a^(2)+b^(2)-ab=c^(2)`, then

Options

A. `min (a,b)lecle max (a,b)`
B. `min(a,b)ltclt max(a,b)`
C. `clemin(a,b)lemax(a,b)`
D. `min(a,b)lemax(a,b)lec`

Answer

B

Explanation

We have,
`a^(2)+b^(2)-ab=c^(2)`
`implies" "(a^(2)+b^(2)-c^(2))/(2ab)=(1)/(2)impliescosC=(1)/(2)implies"C=60^(@)`
`:." "A+B+C=180^(@)impliesA+B=120^(@)`
Following cases arise:
CASE I When A=B
In this case, we have
`A=B=60^(@)=Cimpliesa=b=c`
`implies" "min(a,b)=c=max(a,b)`
CASE II When `AgtB`
In this case, we have
`A+B=120^(@)" and "AgtB`
`:." "Agt60^(@)" and "Blt60^(@)" and "C=60^(@)`
`implies" "agtb,agtc" and "bltc`
`:." "min(a,b)=b" and "max(a,b)=a`
Now, `agtb,agtc" and "bltc`
`implies" "bltclta`
`implies" "min(a,b )ltcltmax(a,b)`
Similarly, when `AltB,` we have
`min(a,b)ltcltmax(a,b)`
Hence, `min(a,b)leclemax(a,b).`