SME
Grade 10 Math · Problem type
The point that splits a segment in a ratio
Use this for “Find the point P on AB, A(−2, 1) and B(10, 7), with AP:PB = 1:2.”
Chart
P is 1/3 of the way from A to B: 1 of the 3 equal pieces (ratio 1 : 2).
P = (−2 + 1/3 × 12, 1 + 1/3 × 6) = (2, 3)
This is an example. Type your own values to start a new problem.
xₚ
=
x₁
+
m
+
(
x₂
−
(
x₁
)
)
yₚ
=
y₁
+
m
+
(
y₂
−
y₁
)
Formulas
xₚ
=
x₁
+
m
m
+
n
×
(
x₂
−
x₁
)
2
=
−2
+
1
1 + 2
×
(10
−
(−2))
yₚ
=
y₁
+
m
m
+
n
×
(
y₂
−
y₁
)
3
=
1
+
1
1 + 2
×
(7
−
1)
Assumptions
•
P is on AB with AP:PB = m:n.
•
Cut AB into m + n equal pieces; P is m pieces from A, the fraction m ÷ (m + n) of the way.
•
Take that fraction of the change across and of the change up, starting from A.
Step-by-step
WE KNOW
x₁ = −2, y₁ = 1, x₂ = 10, y₂ = 7, m = 1, n = 2
FIND
x of P (xₚ), y of P (yₚ)
Step 1 · Find x of P (xₚ)
Use xₚ = x₁ + m ÷ (m + n) × (x₂ − x₁)
Start
at
A
and
go
m
of
the
m
+
n
equal
pieces
of
the
change
in
x.
xₚ
=
−2
+
1
1 + 2
×
(10
−
(−2))
xₚ
=
−2
+
1
3
×
12
xₚ = −2 + 4
xₚ = 2
Step 2 · Find y of P (yₚ)
Use yₚ = y₁ + m ÷ (m + n) × (y₂ − y₁)
Start
at
A
and
go
m
of
the
m
+
n
equal
pieces
of
the
change
in
y.
yₚ
=
1
+
1
1 + 2
×
(7
−
1)
yₚ
=
1
+
1
3
×
6
yₚ = 1 + 2
yₚ = 3
Check
Put the numbers back in. Both sides should match.
2
=
−2
+
1
1 + 2
×
(10
−
(−2))
✓
3
=
1
+
1
1 + 2
×
(7
−
1)
✓
Topic check
See what you know: 5 questions on this topic
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