QQ. If A(5,-4), B(-10,0) are given, find the coordinates of the points which divide seg AB into five congruent parts.
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Step-by-step explanation:
Let the points P(x
1
,y
1
),Q(x
2
,y
2
),R(x
3
,y
3
) and S(x
4
,y
4
) be the points which divide the line segment AB into 5 equal parts.
PB
AP
=
PQ+QR+RS
AP
=
4AP
AP
=
4
1
x
1
=(
1+4
1×0+4×20
)=16
y
1
=(
1+4
1×20+4×10
)=12
P(x
1
,y
1
)=(16,12)
QB
PQ
=
QR+RS+SB
PQ
=
PQ+PQ+PQ
PQ
=
3PQ
PQ
=
3
1
x
2
=(
1+3
1×0+3×16
)=12
y
2
=(
1+3
1×20+3×12
)=14
Q(x
2
,y
2
)=(12,14)
RB
QR
=
RS+SB
QR
=
QR+QR
QR
=
2QR
QR
=
2
1
x
3
=(
1+2
1×0+2×12
)=8
y
3
=(
1+2
1×20+2×14
)=16
R(x
3
,y
3
)=(8.16)
S is the midpoint of RB so, using the midpoint formula
x
4
=
0
8+0
=4
y
4
=
2
16+20
=18
S(x
4
,y
4
)=(4,18)
So, the points
P(x
1
,y
1
)=(16,12)
Q(x
2
,y
2
)=(12,14)
R(x
3
,y
3
)=(8,16)
S(x
4
,y
4
)=(4,18)
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