The vertex A of triangle ABC is (3,-1). The equations of median BE and angular bisector CF are 6x+10y-59 =
0, and x - 4y+10 = 0. Then
Q: The equation of AB must be
Answers
Given : The vertex A of triangle ABC is (3,-1). The equations of median BE and angular bisector CF are 6x+10y–59 = 0, and x - 4y+10 = 0.
To find : The equation of AB
Solution:
A = ( 3 , - 1)
BE is median
=> E is mid point of AC
BE equation 6x + 10 y - 59 = 0
=> E = ( a , (59 - 6a)/10)
CF equation x - 4y + 10 = 0
=> C = ( b , (b + 10)/4)
E is mid point
a = (b + 3)/2 => b = 2a - 3
(59 - 6a)/10 = ( (b + 10)/4 - 1) /2
=> 59 - 6a = 5 (b + 6) /4
=> 236 - 24a = 5 (2a - 3 + 6)
=> 236 - 24a = 10a + 15
=> 34a = 221
=> 2a = 13
=> a = 13/2
b = 10
C = ( 10 , 5)
A = ( 3 , - 1) , C = ( 10 , 5)
Slope of AC = 6/7
Slope of CF = 1/4 ( y = x/4 + 2.5 )
Slope of BC = m
Angle between BC & CF = angle between AC & CF
| m - (1/4) / ( 1 + m(1/4) | = | (1/4) - (6/7) / ( 1 + (1/4)(6/7) |
=> | (4m - 1 )/(4 + m) | = | -17 / (34) |
=> | (4m - 1 )/(4 + m) | = | -1 / 2 |
Case 1
(4m - 1 )/(4 + m) = - 1/2
=> 8m - 2 = -4 - m
=> 9m = -2
=> m = -2/9
Case 2
(4m - 1 )/(4 + m) = 1/2
=> 8m - 2 = 4 + m
=> 7m = 6
=> m = 6/7 ( same as AC )
Hence m = -2/9
BC Equation
y - 5 = (-2/9)(x - 10)
=> 9y - 45 = -2x + 20
=> 2x + 9y = 65
BE = 6x + 10y = 59
=> y = 8 & x = -7/2
B = (-7/2 , 8)
A = ( 3 , - 1)
Slope AB = -18/13
Equation AB y + 1 = (-18/13)(x - 3)
=> 13y + 13 = -18x + 54
=> 18x + 13y = 41
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Answer:
18x + 13y = 41
Step-by-step explanation:
Given : The vertex A of triangle ABC is (3,-1). The equations of median BE and angular bisector CF are 6x+10y–59 = 0, and x - 4y+10 = 0.
To find : The equation of AB
Solution:
A = ( 3 , - 1)
BE is median
=> E is mid point of AC
BE equation 6x + 10 y - 59 = 0
=> E = ( a , (59 - 6a)/10)
CF equation x - 4y + 10 = 0
=> C = ( b , (b + 10)/4)
E is mid point
a = (b + 3)/2 => b = 2a - 3
(59 - 6a)/10 = ( (b + 10)/4 - 1) /2
=> 59 - 6a = 5 (b + 6) /4
=> 236 - 24a = 5 (2a - 3 + 6)
=> 236 - 24a = 10a + 15
=> 34a = 221
=> 2a = 13
=> a = 13/2
b = 10
C = ( 10 , 5)
A = ( 3 , - 1) , C = ( 10 , 5)
Slope of AC = 6/7
Slope of CF = 1/4 ( y = x/4 + 2.5 )
Slope of BC = m
Angle between BC & CF = angle between AC & CF
| m - (1/4) / ( 1 + m(1/4) | = | (1/4) - (6/7) / ( 1 + (1/4)(6/7) |
=> | (4m - 1 )/(4 + m) | = | -17 / (34) |
=> | (4m - 1 )/(4 + m) | = | -1 / 2 |
Case 1
(4m - 1 )/(4 + m) = - 1/2
=> 8m - 2 = -4 - m
=> 9m = -2
=> m = -2/9
Case 2
(4m - 1 )/(4 + m) = 1/2
=> 8m - 2 = 4 + m
=> 7m = 6
=> m = 6/7 ( same as AC )
Hence m = -2/9
BC Equation
y - 5 = (-2/9)(x - 10)
=> 9y - 45 = -2x + 20
=> 2x + 9y = 65
BE = 6x + 10y = 59
=> y = 8 & x = -7/2
B = (-7/2 , 8)
A = ( 3 , - 1)
Slope AB = -18/13
Equation AB y + 1 = (-18/13)(x - 3)
=> 13y + 13 = -18x + 54
=> 18x + 13y = 41