Coordinate Geometry Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem step by step using the Distance Formula and Midpoint Formula as provided.

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🔹 Distance Formula:

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

We'll apply this to the first six problems.

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#### 1. (-3, 5) and (-7, -8)
$$
d = \sqrt{(-7 - (-3))^2 + (-8 - 5)^2} = \sqrt{(-4)^2 + (-13)^2} = \sqrt{16 + 169} = \sqrt{185}
$$
✔ Answer: $ \sqrt{185} \approx 13.60 $

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#### 2. (-5, 8) and (-4, 1)
$$
d = \sqrt{(-4 - (-5))^2 + (1 - 8)^2} = \sqrt{(1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2}
$$
✔ Answer: $ 5\sqrt{2} \approx 7.07 $

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#### 3. (5, 4) and (3, 8)
$$
d = \sqrt{(3 - 5)^2 + (8 - 4)^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
$$
✔ Answer: $ 2\sqrt{5} \approx 4.47 $

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#### 4. (0, -6) and (-7, 1)
$$
d = \sqrt{(-7 - 0)^2 + (1 - (-6))^2} = \sqrt{(-7)^2 + (7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}
$$
✔ Answer: $ 7\sqrt{2} \approx 9.90 $

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#### 5. (2, -3) and (8, -5)
$$
d = \sqrt{(8 - 2)^2 + (-5 - (-3))^2} = \sqrt{(6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}
$$
✔ Answer: $ 2\sqrt{10} \approx 6.32 $

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#### 6. (-4, -2) and (7, -3)
$$
d = \sqrt{(7 - (-4))^2 + (-3 - (-2))^2} = \sqrt{(11)^2 + (-1)^2} = \sqrt{121 + 1} = \sqrt{122}
$$
✔ Answer: $ \sqrt{122} \approx 11.05 $

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🔹 Midpoint Formula:

$$
(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

Now solve the midpoint problems.

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#### 1. (8, 0) and (-9, 1)
$$
x = \frac{8 + (-9)}{2} = \frac{-1}{2} = -0.5,\quad y = \frac{0 + 1}{2} = \frac{1}{2} = 0.5
$$
✔ Answer: $ \left( -\frac{1}{2}, \frac{1}{2} \right) $

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#### 2. (-9, -10) and (4, -9)
$$
x = \frac{-9 + 4}{2} = \frac{-5}{2} = -2.5,\quad y = \frac{-10 + (-9)}{2} = \frac{-19}{2} = -9.5
$$
✔ Answer: $ \left( -\frac{5}{2}, -\frac{19}{2} \right) $

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#### 3. (4, -3) and (2, 9)
$$
x = \frac{4 + 2}{2} = \frac{6}{2} = 3,\quad y = \frac{-3 + 9}{2} = \frac{6}{2} = 3
$$
✔ Answer: $ (3, 3) $

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#### 4. (8, 7) and (-8, 3)
$$
x = \frac{8 + (-8)}{2} = \frac{0}{2} = 0,\quad y = \frac{7 + 3}{2} = \frac{10}{2} = 5
$$
✔ Answer: $ (0, 5) $

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#### 5. (7, -8) and (-2, 8)
$$
x = \frac{7 + (-2)}{2} = \frac{5}{2} = 2.5,\quad y = \frac{-8 + 8}{2} = \frac{0}{2} = 0
$$
✔ Answer: $ \left( \frac{5}{2}, 0 \right) $

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#### 6. (-4, 7) and (3, -1)
$$
x = \frac{-4 + 3}{2} = \frac{-1}{2} = -0.5,\quad y = \frac{7 + (-1)}{2} = \frac{6}{2} = 3
$$
✔ Answer: $ \left( -\frac{1}{2}, 3 \right) $

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✔ Final Answers Summary:

#### Distance Problems:
1. $ \sqrt{185} $
2. $ 5\sqrt{2} $
3. $ 2\sqrt{5} $
4. $ 7\sqrt{2} $
5. $ 2\sqrt{10} $
6. $ \sqrt{122} $

#### Midpoint Problems:
1. $ \left( -\frac{1}{2}, \frac{1}{2} \right) $
2. $ \left( -\frac{5}{2}, -\frac{19}{2} \right) $
3. $ (3, 3) $
4. $ (0, 5) $
5. $ \left( \frac{5}{2}, 0 \right) $
6. $ \left( -\frac{1}{2}, 3 \right) $

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