Partitioning a Segment in a Ratio
Aligned to G-GPE.B.6 — Common Core State Standards for Mathematics.
What this lesson teaches
To find the point that divides a directed segment from A to B in the ratio m:n, move a fraction m/(m+n) of the way from A to B. The coordinates are A plus that fraction of the change in x and y. The midpoint is the special case with ratio 1:1.
Worked examples
Example 1
Find the point that partitions A(2, 3) to B(10, 11) in the ratio 1:3. The fraction from A is 1/(1+3) = 1/4. Change in x is 10 - 2 = 8; change in y is 11 - 3 = 8. Move 1/4 of each: x = 2 + (1/4)(8) = 2 + 2 = 4; y = 3 + (1/4)(8) = 3 + 2 = 5. The point is (4, 5).
Example 2
Find the midpoint of A(-3, 7) and B(5, 1). The midpoint is the 1:1 partition, so the fraction from A is 1/2. Change in x is 5 - (-3) = 8 and change in y is 1 - 7 = -6. Move half of each: x = -3 + 4 = 1, y = 7 - 3 = 4. The midpoint is (1, 4) — which is also just the average of the coordinates.
Example 3
Common mistake: turning the ratio 2:3 into the fraction 2/3. The ratio names the two PIECES, so the whole is 2 + 3 = 5 parts and the fraction from the first point is 2/5, not 2/3. Partitioning A(0,0) to B(10,0) in 2:3 gives x = 0 + (2/5)(10) = 4, leaving pieces of 4 and 6 — a 2:3 split. Using 2/3 would give 6.67 and pieces of 6.67 and 3.33, which is 2:1.
Practice questions
- Find the midpoint of A(1, 2) and B(7, 10).
- Find the point 2/5 of the way from A(0, 0) to B(10, 15).
- Partition the segment from A(-4, 1) to B(8, 7) in the ratio 3:1.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Dividir un segmento en una razón
Para hallar el punto que divide un segmento dirigido de A a B en la razón m:n, avanza una fracción m/(m+n) del camino de A a B. Las coordenadas son A más esa fracción del cambio en x y en y. El punto medio es el caso especial con razón 1:1.
Ejemplo 1: Halla el punto que divide A(2, 3) a B(10, 11) en la razón 1:3. La fracción desde A es 1/(1+3) = 1/4. El cambio en x es 10 - 2 = 8; el cambio en y es 11 - 3 = 8. Avanza 1/4 de cada uno: x = 2 + (1/4)(8) = 2 + 2 = 4; y = 3 + (1/4)(8) = 3 + 2 = 5. El punto es (4, 5).
Ejemplo 2: Halla el punto medio de A(-3, 7) y B(5, 1). El punto medio es la partición 1:1, así que la fracción desde A es 1/2. El cambio en x es 5 - (-3) = 8 y el cambio en y es 1 - 7 = -6. Avanza la mitad de cada uno: x = -3 + 4 = 1, y = 7 - 3 = 4. El punto medio es (1, 4) — que además es simplemente el promedio de las coordenadas.
Ejemplo 3: Error común: convertir la razón 2:3 en la fracción 2/3. La razón nombra las dos PARTES, así que el total es 2 + 3 = 5 partes y la fracción desde el primer punto es 2/5, no 2/3. Partir A(0,0) a B(10,0) en 2:3 da x = 0 + (2/5)(10) = 4, dejando partes de 4 y 6 — una división 2:3. Usar 2/3 daría 6.67 y partes de 6.67 y 3.33, que es 2:1.
More Expressing Geometric Properties with Equations lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required