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

Let's solve each problem step-by-step using the section formula for partitioning a line segment.

---

🔷 Section Formula (Internal Division):

If a point $ P $ divides the line segment joining points $ A(x_1, y_1) $ and $ B(x_2, y_2) $ in the ratio $ m:n $, then the coordinates of $ P $ are:

$$
P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
$$

We'll use this formula throughout.

---

## ✔ Problem 1:
D(5, 0), E(2, -8), ratio 1:4

Here, $ D = (5, 0) $, $ E = (2, -8) $, ratio $ m:n = 1:4 $

$$
x = \frac{1 \cdot 2 + 4 \cdot 5}{1+4} = \frac{2 + 20}{5} = \frac{22}{5} = 4.4
$$
$$
y = \frac{1 \cdot (-8) + 4 \cdot 0}{5} = \frac{-8}{5} = -1.6
$$

✔ Answer: $ (4.4, -1.6) $

---

## ✔ Problem 2:
D(-3, 7), E(-2, 5), ratio 2:5

$ m = 2, n = 5 $

$$
x = \frac{2 \cdot (-2) + 5 \cdot (-3)}{2+5} = \frac{-4 -15}{7} = \frac{-19}{7} \approx -2.7
$$
$$
y = \frac{2 \cdot 5 + 5 \cdot 7}{7} = \frac{10 + 35}{7} = \frac{45}{7} \approx 6.4
$$

✔ Answer: $ (-2.7, 6.4) $

---

## ✔ Problem 3:
D(10, -13), E(14, 18), ratio 1:2

$ m = 1, n = 2 $

$$
x = \frac{1 \cdot 14 + 2 \cdot 10}{1+2} = \frac{14 + 20}{3} = \frac{34}{3} \approx 11.3
$$
$$
y = \frac{1 \cdot 18 + 2 \cdot (-13)}{3} = \frac{18 - 26}{3} = \frac{-8}{3} \approx -2.7
$$

✔ Answer: $ (11.3, -2.7) $

---

## ✔ Problem 4:
D(-20, -22), E(0, 14), ratio 2:3

$ m = 2, n = 3 $

$$
x = \frac{2 \cdot 0 + 3 \cdot (-20)}{2+3} = \frac{0 - 60}{5} = -12.0
$$
$$
y = \frac{2 \cdot 14 + 3 \cdot (-22)}{5} = \frac{28 - 66}{5} = \frac{-38}{5} = -7.6
$$

✔ Answer: $ (-12.0, -7.6) $

---

## ✔ Problem 5:
Find the point that is 1/3 the way from A to B, where $ A(0, 10) $, $ B(-16, 14) $

This means the ratio is $ 1:2 $ (since 1 part out of 3 total parts)

So $ m = 1, n = 2 $

$$
x = \frac{1 \cdot (-16) + 2 \cdot 0}{1+2} = \frac{-16}{3} \approx -5.3
$$
$$
y = \frac{1 \cdot 14 + 2 \cdot 10}{3} = \frac{14 + 20}{3} = \frac{34}{3} \approx 11.3
$$

✔ Answer: $ (-5.3, 11.3) $

---

## ✔ Problem 6:
Find the point that is 3/4 the way from A to B, where $ A(22, 19) $, $ B(-16, 7) $

This means $ m = 3, n = 1 $ (because 3 parts out of 4)

$$
x = \frac{3 \cdot (-16) + 1 \cdot 22}{3+1} = \frac{-48 + 22}{4} = \frac{-26}{4} = -6.5
$$
$$
y = \frac{3 \cdot 7 + 1 \cdot 19}{4} = \frac{21 + 19}{4} = \frac{40}{4} = 10.0
$$

✔ Answer: $ (-6.5, 10.0) $

---

## ✔ Problem 7:
Find the point that is 2/5 the way from A to B, where $ A(-2, 8) $, $ B(13, 9) $

Ratio = $ 2:3 $

$$
x = \frac{2 \cdot 13 + 3 \cdot (-2)}{2+3} = \frac{26 - 6}{5} = \frac{20}{5} = 4.0
$$
$$
y = \frac{2 \cdot 9 + 3 \cdot 8}{5} = \frac{18 + 24}{5} = \frac{42}{5} = 8.4
$$

✔ Answer: $ (4.0, 8.4) $

---

## ✔ Problem 8:
Find the point that partitions $ \overline{AB} $ with a ratio of 1:3

From graph:
- Point A is at $ (-4, -2) $
- Point B is at $ (2, 6) $

Check: From grid:
- A is at (-4, -2)
- B is at (2, 6)

Use section formula with $ m:n = 1:3 $

$$
x = \frac{1 \cdot 2 + 3 \cdot (-4)}{1+3} = \frac{2 - 12}{4} = \frac{-10}{4} = -2.5
$$
$$
y = \frac{1 \cdot 6 + 3 \cdot (-2)}{4} = \frac{6 - 6}{4} = \frac{0}{4} = 0.0
$$

✔ Answer: $ (-2.5, 0.0) $

---

## ✔ Problem 9:
Find the point that is 3/5 the distance from A to B

From graph:
- A is at $ (2, -4) $
- B is at $ (-4, 2) $

So $ m:n = 3:2 $ (3 parts toward B, 2 left)

$$
x = \frac{3 \cdot (-4) + 2 \cdot 2}{3+2} = \frac{-12 + 4}{5} = \frac{-8}{5} = -1.6
$$
$$
y = \frac{3 \cdot 2 + 2 \cdot (-4)}{5} = \frac{6 - 8}{5} = \frac{-2}{5} = -0.4
$$

✔ Answer: $ (-1.6, -0.4) $

---

✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ (4.4, -1.6) $ |
| 2 | $ (-2.7, 6.4) $ |
| 3 | $ (11.3, -2.7) $ |
| 4 | $ (-12.0, -7.6) $ |
| 5 | $ (-5.3, 11.3) $ |
| 6 | $ (-6.5, 10.0) $ |
| 7 | $ (4.0, 8.4) $ |
| 8 | $ (-2.5, 0.0) $ |
| 9 | $ (-1.6, -0.4) $ |

---

📌 Explanation Recap:

To find a point that divides a segment in a given ratio or fraction:
- Use the section formula: $ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $
- For "k/n the way from A to B", set ratio $ k:(n-k) $
- Always verify coordinates from graphs when provided.
- Round to tenths place as instructed.

Let me know if you'd like this in a printable format!