Coordinate Geometry Notes and Worksheets - Lindsay Bowden - Free Printable

Absolutely! Let's solve each problem step by step using the Distance Formula and the Midpoint Formula as given in your worksheet.

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## ✔ PART 1: DISTANCE FORMULA

The distance formula is:

> d = √[(x₂ - x₁)² + (y₂ - y₁)²]

We’ll plug in the coordinates for each pair of points.

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1. Points: (-3, 5) and (-7, -8)

Let (x₁, y₁) = (-3, 5), (x₂, y₂) = (-7, -8)

d = √[(-7 - (-3))² + (-8 - 5)²]
= √[(-4)² + (-13)²]
= √[16 + 169]
= √185
→ Answer: √185 (or approximately 13.60 if decimal needed)

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2. Points: (-5, 8) and (-4, 1)

(x₁, y₁) = (-5, 8), (x₂, y₂) = (-4, 1)

d = √[(-4 - (-5))² + (1 - 8)²]
= √[(1)² + (-7)²]
= √[1 + 49]
= √50 = √(25×2) = 5√2

→ Answer: 5√2 (≈7.07)

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3. Points: (5, 4) and (3, 8)

(x₁, y₁) = (5, 4), (x₂, y₂) = (3, 8)

d = √[(3 - 5)² + (8 - 4)²]
= √[(-2)² + (4)²]
= √[4 + 16]
= √20 = √(4×5) = 2√5

→ Answer: 2√5 (≈4.47)

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4. Points: (0, -6) and (-7, 1)

(x₁, y₁) = (0, -6), (x₂, y₂) = (-7, 1)

d = √[(-7 - 0)² + (1 - (-6))²]
= √[(-7)² + (7)²]
= √[49 + 49]
= √98 = √(49×2) = 7√2

→ Answer: 7√2 (≈9.90)

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5. Points: (2, -3) and (8, -5)

(x₁, y₁) = (2, -3), (x₂, y₂) = (8, -5)

d = √[(8 - 2)² + (-5 - (-3))²]
= √[(6)² + (-2)²]
= √[36 + 4]
= √40 = √(4×10) = 2√10

→ Answer: 2√10 (≈6.32)

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6. Points: (-4, -2) and (7, -3)

(x₁, y₁) = (-4, -2), (x₂, y₂) = (7, -3)

d = √[(7 - (-4))² + (-3 - (-2))²]
= √[(11)² + (-1)²]
= √[121 + 1]
= √122

→ Answer: √122 (≈11.05)

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## ✔ PART 2: MIDPOINT FORMULA

The midpoint formula is:

> (x, y) = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

We’ll average the x-coordinates and y-coordinates separately.

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1. Points: (8, 0) and (-9, 1)

x = (8 + (-9))/2 = (-1)/2 = -0.5
y = (0 + 1)/2 = 1/2 = 0.5

→ Answer: (-0.5, 0.5) or (-1/2, 1/2)

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2. Points: (-9, -10) and (4, -9)

x = (-9 + 4)/2 = (-5)/2 = -2.5
y = (-10 + (-9))/2 = (-19)/2 = -9.5

→ Answer: (-2.5, -9.5) or (-5/2, -19/2)

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3. Points: (4, -3) and (2, 9)

x = (4 + 2)/2 = 6/2 = 3
y = (-3 + 9)/2 = 6/2 = 3

→ Answer: (3, 3)

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4. Points: (8, 7) and (-8, 3)

x = (8 + (-8))/2 = 0/2 = 0
y = (7 + 3)/2 = 10/2 = 5

→ Answer: (0, 5)

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5. Points: (7, -8) and (-2, 8)

x = (7 + (-2))/2 = 5/2 = 2.5
y = (-8 + 8)/2 = 0/2 = 0

→ Answer: (2.5, 0) or (5/2, 0)

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6. Points: (-4, 7) and (3, -1)

x = (-4 + 3)/2 = (-1)/2 = -0.5
y = (7 + (-1))/2 = 6/2 = 3

→ Answer: (-0.5, 3) or (-1/2, 3)

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## 📝 FINAL ANSWERS SUMMARY:

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🔹 Distance Formula Answers:

1. √185
2. 5√2
3. 2√5
4. 7√2
5. 2√10
6. √122

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🔹 Midpoint Formula Answers:

1. (-0.5, 0.5) or (-1/2, 1/2)
2. (-2.5, -9.5) or (-5/2, -19/2)
3. (3, 3)
4. (0, 5)
5. (2.5, 0) or (5/2, 0)
6. (-0.5, 3) or (-1/2, 3)

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✔ You can now fill these into your worksheet!

Let me know if you’d like to see any step explained further or want to convert radicals to decimals!