A(2,5), B(-1,2) and C(5,8) are the vertices of the triangle ABC. M is the point on AB such that AM:MB =1:2. Find the coordinates of point M, hence find the equation of the line passing through the points C and M
Answers
Answer:
M is a point which divides the line segment AB internally in the ratio 1:2 because it is on AB and not on the extended portion of AB.
To find the coodrinates of a point (X,Y) which divides a line segment joining (x
A
,y
A
) and (x
B
,y
B
) internally in the ratio m:n, we use the formula:
X=
m+n
mx
B
+nx
A
and
Y=
m+n
my
B
+ny
A
So the point M can be found with the formula:
x
M
=
3
2x
A
+x
B
=
3
2×2−1
=1 and
y
M
=
3
2y
A
+y
B
=
3
2×5+2
=4
Thus, the coordinates of the point M are (1,4)
The equation of the line passing through points C and M can be obtained using the two-point form of line equation as:
x−x
C
y−y
C
=
x
M
−x
C
y
M
−y
C
x−5
y−8
=
1−5
4−8
=1
⇒x−y+3=0
Thus the equation of the line passing through the points C and M is x−y+3=0
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