Three points A(1, 3), B(5, 0), C(3, 8) are the vertices of triangle AB

Question

Three points  A(1, 3), B(5, 0), C(3, 8)  are the vertices of triangle  ABC . If the line  y =mx - m + 3  divides triangle  ABC  evenly, what is  m ?

A.  -1  
 
B.  \frac{1}{3} 
 
C.  \frac{5}{6}  
 
D.  \frac{7}{8} 
 
E.  1

MathRevolution · Accepted Answer

=> 1.19.png y = mx – m + 3 = m(x - 1) + 3 passes through A(1, 3). Since the line divides the triangle ABC evenly, it passes through the mid-point (4, 4) of BC. Then the slope is \frac{(4 - 3) }{ (4 - 1)} = \frac{1}{3}. Thus we have m = \frac{1}{3}. Therefore, B is the answer. Answer: B

yashwardhan · Answer

Can anyone help me with the solution?

firas92 · Answer

Note that  m  is the slope of the required line

The dividing line would join vertrex A with the mid point of BC

Mid point of BC = ( \frac{3+5}{2},\frac{0+8}{2} ) =  (4,4)

Slope of line joining (1,3) and (4,4),  m= \frac{4-3}{4-1} = \frac{1}{3}

Answer is  (B)

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yashwardhan · Answer

firas92  why can't I join vertex C with mid pt. of AB or Vertex B with mid pt. of AC?

firas92 · Answer

You can but I don't find those options in the answer choices. I would like to see the OE as well

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yashwardhan · Answer

firas92  Do you think it has something to do with the equation y=mx−m+3 ?
When I tried taking out 3 different slopes(m) I got fractions in all except one of them, where the slope was -4. But I still dont know how any of that factors to get the condition of joining vertex A with the mid pt. of BC!

Can,Will · Answer

@mathsrevolution pls expain this as mentioned by 
 firas92  why can't I join vertex C with mid pt. of AB or Vertex B with mid pt. of AC?

Can,Will · Answer

Bunuel  
 pls explain this

Gragnor · Answer

I believe it has got to do with the centroid of the triangle which would be ((1+5+3)/3,(3+0+8)/3) i.e. (3,11/3)

Putting that into the equation, you get - 11/3 = 3m - m +3 ==> 2/3 = 2m

AnuK2222 · Answer

Answer is B, m=1/3
The eq of y=mx-m+3 can be rewritten as y=(x-1)m+3... this indicates that the line is has positive slop and positive y intercept... based on this draw the diagram and we see that since the line divides the triangle evenly it will meet the line BC at BC's midpoint ie (4,4)
substituting (4,4) in y=(x-1)m+3 we get m=3

crimson_noise · Answer

y = mx - m + 3
First notice something important.
When x = 1:
y = m(1) - m + 3 = 3
So the line always passes through A(1,3) regardless of m.

Find midpoint of BC
B(5,0), C(3,8)
Midpoint:
((5+3)/2), (0+8)/2)=(4,4)

Find slope of line from A to midpoint

Slope:
m = (4 - 3)/(4 - 1)=1/3

Three points A(1, 3), B(5, 0), C(3, 8) are the vertices of triangle AB

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Three points A(1, 3), B(5, 0), C(3, 8) are the vertices of triangle AB

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