Coordinate Distance

Grade 6 · geometry · 100 practice problems · read aloud

🔊 Listen to this explanation

📏 Finding the Distance Between Coordinates

What is Coordinate Distance?

It's a way to measure how far apart two points are on a coordinate plane. This is super useful for map reading, planning routes, and even in game design!

How to Find the Distance

When two points share an X-coordinate or a Y-coordinate, finding the distance is easy! You just count the spaces between them.

  1. Step 1: Identify the two points (e.g., A(2, 5) and B(2, 1)).
  2. Step 2: Find the coordinates that are different.
  3. Step 3: Subtract the smaller number from the larger number.
  4. Step 4: The answer is the distance!

Visual Examples

Example 1: Vertical Distance

Points: C(3, 4) and D(3, 1)

The X-coordinates are the same (3), so we find the difference in the Y-coordinates: |4 - 1| = 3. The distance is 3 units.

Example 2: Horizontal Distance

Points: E(1, 2) and F(5, 2)

The Y-coordinates are the same (2), so we find the difference in the X-coordinates: |5 - 1| = 4. The distance is 4 units.

⚠️ Common Mistakes

  • Subtracting in the wrong order: Always subtract the smaller number from the larger number, or use absolute value (| |) so your answer is always positive. Distance can't be negative!
  • Mixing up X and Y: Make sure you are subtracting the coordinates that are different. If the X's are the same, subtract the Y's. If the Y's are the same, subtract the X's.

💡 Tips & Tricks

  • Memory Aid: "Same X? Find the Y! Same Y? Find the X!"
  • Draw it! Plot the points on graph paper. Counting the squares between them is a great way to check your work.

How to Practice

Try these on graph paper:

  • Find the distance between (6, 3) and (6, 8).
  • Find the distance between (2, 5) and (7, 5).
  • Create your own points that are 5 units apart vertically.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 √(169) + |−15| = ?

Hint: First find the principal square root of the perfect square, then evaluate the absolute value of the negative number, and finally add the results.

Show the answer

Answer: 28

  1. Understand the expression. We need to evaluate √(169) + |−15|.
  2. Evaluate the square root. √(169) means the positive square root of 169. Since 13 × 13 = 169, we have √(169) = 13.
  3. Evaluate the absolute value. |−15| means the distance from −15 to 0 on the number line, which is 15. So |−15| = 15.
  4. Add the results. 13 + 15 = 28.
  5. Final answer. The result is 28.

2 (-12) + 25 - 8 = ?

Hint: Remember to work with negative numbers carefully, performing operations from left to right.

Show the answer

Answer: 5

  1. Start with the first two numbers: (-12) + 25 Think of this as 25 minus 12, because adding a negative is the same as subtracting a positive. 25 - 12 = 13 So, (-12) + 25 = 13
  2. Now take the result from Step 1 and subtract 8: 13 - 8 = 5
  3. Final answer: 5 So, (-12) + 25 - 8 = 5

Let's solve the problem step by step. We have: (-12) + 25 - 8

3 √(144) + 3² - 5 = ?

Hint: Remember to evaluate each mathematical operation in the correct order, starting with any exponents or roots.

Show the answer

Answer: 16

  1. Evaluate the square root √(144) means the positive square root of 144. Since 12 × 12 = 144, we have √(144) = 12. So the expression becomes: 12 + 3² − 5.
  2. Evaluate the exponent 3² means 3 × 3 = 9. Now the expression is: 12 + 9 − 5.
  3. Perform addition and subtraction from left to right First, 12 + 9 = 21. Then, 21 − 5 = 16. Final Answer: 16

Let's solve step by step.

4 (-8)² + 12 ÷ (-3) = ?

Hint: Remember to follow the order of operations and pay attention to negative signs when working with exponents and division.

Show the answer

Answer: 60

  1. Evaluate (-8)² This means (-8) × (-8) = 64. Reason: Squaring a negative number gives a positive result.
  2. Evaluate 12 ÷ (-3) 12 divided by -3 equals -4. Reason: A positive divided by a negative is negative.
  3. Add the results from Step 1 and Step 2 64 + (-4) = 64 - 4 = 60. Final answer: 60

Let's solve step-by-step.

5 (-8)² - 12 ÷ (-3) = ?

Hint: Remember the order of operations and how negative numbers behave with exponents and division.

Show the answer

Answer: 68

  1. Evaluate the exponent first: (-8)² = (-8) × (-8) = 64
  2. Evaluate the division: 12 ÷ (-3) = -4
  3. Substitute the results back into the expression: 64 - (-4)
  4. Subtracting a negative is the same as adding a positive: 64 + 4 = 68

The answer is 68.

6 √(144) + 3² - |−5| = ?

Hint: Remember to evaluate each part separately: square root, exponent, and absolute value before combining them.

Show the answer

Answer: 16

  1. Evaluate the square root √(144) = 12, because 12 × 12 = 144. So the expression becomes: 12 + 3² − |−5|
  2. Evaluate the exponent 3² = 3 × 3 = 9 Now we have: 12 + 9 − |−5|
  3. Evaluate the absolute value |−5| = 5, because absolute value means the distance from zero, ignoring sign. Now we have: 12 + 9 − 5
  4. Perform addition and subtraction from left to right 12 + 9 = 21 21 − 5 = 16 Final answer: 16

Let's solve step-by-step.

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