Partitioning a Line Segment Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve the problem step by step, starting with filling in the notes section, then solving each example.

---

🔹 FILL IN THE BLANKS (Notes Section)

> *Partitioning a line segment means dividing the line segment into parts. The formula is used when we want to find a point on the line segment other than the midpoint.*

> The ratio in which the line segment needs to be divided will be given as a:b.

✔ So the completed notes are:

- Partitioning a line segment means dividing the line segment into parts.
- The formula is used when we want to find a point on the line segment other than the midpoint.
- The ratio in which the line segment needs to be divided will be given as a:b.

The formula:
$$
(x, y) = \left( \frac{bx_1 + ax_2}{b + a}, \frac{by_1 + ay_2}{b + a} \right)
$$
This formula finds a point that divides the segment from point $ A(x_1, y_1) $ to $ B(x_2, y_2) $ in the ratio $ a:b $, where $ a $ is the part toward $ B $, and $ b $ is the part toward $ A $. (Note: This is sometimes written differently depending on convention — make sure the order matches the direction.)

We’ll use this formula carefully for each problem.

---

## ✔ Example 1:
The line segment $\overline{AB}$ has endpoints $A(-5, 3)$ and $B(-1, -5)$. Find the point that partitions the line segment in a ratio of 1:3.

Step 1: Identify values

- $ A(x_1, y_1) = (-5, 3) $
- $ B(x_2, y_2) = (-1, -5) $
- Ratio $ a:b = 1:3 $

So:
- $ a = 1 $, $ b = 3 $

Use the formula:
$$
(x, y) = \left( \frac{b x_1 + a x_2}{a + b}, \frac{b y_1 + a y_2}{a + b} \right)
$$

Plug in:
$$
x = \frac{3(-5) + 1(-1)}{1 + 3} = \frac{-15 - 1}{4} = \frac{-16}{4} = -4
$$
$$
y = \frac{3(3) + 1(-5)}{4} = \frac{9 - 5}{4} = \frac{4}{4} = 1
$$

✔ Answer: $(-4, 1)$

---

## ✔ Example 2:
Find the point that partitions the line segment $\overline{CD}$ in a ratio of 2:1.

We’re given a graph with points $C$ and $D$, but no coordinates. Let’s read the grid.

From the graph:
- Point $C$ is at $(-2, -2)$
- Point $D$ is at $(4, 4)$

Check:
- From $C(-2,-2)$ to $D(4,4)$, it goes up and right.

Now, ratio = 2:1 → $ a:b = 2:1 $

So:
- $ a = 2 $, $ b = 1 $
- $ C(x_1, y_1) = (-2, -2) $
- $ D(x_2, y_2) = (4, 4) $

Apply the formula:
$$
x = \frac{1(-2) + 2(4)}{2 + 1} = \frac{-2 + 8}{3} = \frac{6}{3} = 2
$$
$$
y = \frac{1(-2) + 2(4)}{3} = \frac{-2 + 8}{3} = \frac{6}{3} = 2
$$

✔ Answer: $(2, 2)$

---

## ✔ Example 3:
The line segment $\overline{JK}$ has endpoints $J(-2, -3)$ and $K(2, 5)$. Find the point that is $ \frac{2}{3} $ the distance from $J$ to $K$.

This means: the point is $ \frac{2}{3} $ of the way from $J$ to $K$, so the ratio is:
- Distance from $J$ to point : Distance from point to $K$ = $2:1$

So the ratio $ a:b = 2:1 $, where $ a $ is the part toward $K$, $ b $ is toward $J$

Thus:
- $ a = 2 $, $ b = 1 $
- $ J(x_1, y_1) = (-2, -3) $
- $ K(x_2, y_2) = (2, 5) $

$$
x = \frac{1(-2) + 2(2)}{2+1} = \frac{-2 + 4}{3} = \frac{2}{3}
$$
$$
y = \frac{1(-3) + 2(5)}{3} = \frac{-3 + 10}{3} = \frac{7}{3}
$$

✔ Answer: $ \left( \frac{2}{3}, \frac{7}{3} \right) $

---

## ✔ Example 4:
The line segment $\overline{PQ}$ has endpoints $P(-3, 4)$, $Q(7, 8)$. Find the point that is $ \frac{1}{5} $ the distance from $P$ to $Q$.

So the point is $ \frac{1}{5} $ of the way from $P$ to $Q$, meaning:
- Ratio of $P$ to point : point to $Q$ = $1:4$

So $ a:b = 1:4 $

- $ a = 1 $, $ b = 4 $
- $ P(x_1, y_1) = (-3, 4) $
- $ Q(x_2, y_2) = (7, 8) $

$$
x = \frac{4(-3) + 1(7)}{1 + 4} = \frac{-12 + 7}{5} = \frac{-5}{5} = -1
$$
$$
y = \frac{4(4) + 1(8)}{5} = \frac{16 + 8}{5} = \frac{24}{5} = 4.8
$$

✔ Answer: $(-1, 4.8)$ or $(-1, \frac{24}{5})$

---

✔ Final Answers Summary:

1. $(-4, 1)$
2. $(2, 2)$
3. $\left( \frac{2}{3}, \frac{7}{3} \right)$
4. $(-1, \frac{24}{5})$ or $(-1, 4.8)$

---

📌 Key Concept Recap:

- Partitioning a line segment means dividing it into parts in a given ratio.
- Use the section formula:
$$
(x, y) = \left( \frac{b x_1 + a x_2}{a + b}, \frac{b y_1 + a y_2}{a + b} \right)
$$
where the ratio is $ a:b $, and you're going from point $A(x_1, y_1)$ to $B(x_2, y_2)$, with $a$ being the part toward $B$, and $b$ toward $A$.
- For "distance from A to B", if it's $ \frac{m}{n} $ of the way, then ratio is $ m:(n-m) $

Let me know if you'd like this formatted as a filled-in worksheet!