Question

A(a +1, a – 1), B(a² +1, a²- 1)and C(a³ +1, a³ - 1) are given points D(11,9) is the mid point of AB and E(41,39) is the mid point of BC.If F is the mid point of AC then (BF) is equal to​

Answer 1

Here A(a +1, a – 1), B(a² +1, a²- 1)and C(a³ +1, a³ - 1) are the given points of the Triangle ABC. We shall determine BF = 18√2 ≈ 25.46 units

• D(11,9) is the mid point of AB
• E(41,39) is the mid point of BC
• F is the mid point of AC
• Now the D = (  ((a+1)+(a²+1))/2  , ((a-1)+(a²-1))/2  )

                          =  (  (a+a²+2)/2 , (a+a²-2)/2  )

• So, (a² + a + 2)/2 = 11

     or,     a² + a + 2 = 22

     or,     a² + a -20 = 0

     or,     (a+5)(a-4) =  0

     or,     a = 4  (∵a>0)

• Now the mid point of AC is F =  ( ( (a+1) + (a³ + 1) )/2 , ( (a-1) + (a³- 1)) /2 )

                                                    = ( (a³ + a + 2)/2 , (a³ + a -2)/2 )

• Now putting a = 4 we get,
• F = (35, 33) and B = (17, 15)
• So, BF = √(35-17)² + (33-15)² = 18√2 ≈ 25.46 units