Question 2
Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.
📋 Given Information
- First point (Origin): \( O(0, 0) \) where \( x_1 = 0, y_1 = 0 \)
- Second point: \( P(36, 15) \) where \( x_2 = 36, y_2 = 15 \)
- Context: Towns A and B are located at these coordinates on a map
🎯 To Find
Distance between the origin and point (36, 15), which represents the distance between towns A and B.
📐 Formula
📝 Step-by-Step Solution
Origin: \( x_1 = 0, y_1 = 0 \)
Point P: \( x_2 = 36, y_2 = 15 \)
\( 36^2 = 1296 \)
\( 15^2 = 225 \)
\[ OP = \sqrt{1296 + 225} \]🏙️ Real-World Application: Towns A and B
Context from Section 7.2: In the textbook example, towns A and B are located on a coordinate grid where each unit represents 100 meters.
Calculation:
Distance on grid = 39 units
Actual distance = 39 × 100 meters = 3900 meters = 3.9 km
Conclusion: The straight-line distance between Town A and Town B is 3.9 kilometers.
💡 Alternative Method: Using Pythagoras Theorem
Since the origin and point (36, 15) form a right triangle with the axes:
Step 1: Identify the triangle sides
- Base (along x-axis) = 36 units
- Height (along y-axis) = 15 units
- Hypotenuse = distance OP
Step 2: Apply Pythagoras theorem
\[ OP^2 = 36^2 + 15^2 \] \[ OP^2 = 1296 + 225 = 1521 \]Step 3: Find the square root
\[ OP = \sqrt{1521} = 39 \text{ units} \]This confirms our answer!
🔍 Verification: Checking if 39 is correct
Let’s verify that \( \sqrt{1521} = 39 \):
Method 1: Direct multiplication
\[ 39 \times 39 = 1521 \, ✓ \]Method 2: Prime factorization
\( 1521 = 3 \times 507 = 3 \times 3 \times 169 = 3^2 \times 13^2 \)
\( \sqrt{1521} = \sqrt{3^2 \times 13^2} = 3 \times 13 = 39 \, ✓ \)
Interesting fact: 39 is a composite number (3 × 13), and 1521 is a perfect square!
⚠️ Common Mistakes
❌ Mistake 1: Adding before squaring
Incorrect: \( \sqrt{(36 + 15)^2} = \sqrt{51^2} = 51 \)
✓ Correct: \( \sqrt{36^2 + 15^2} = \sqrt{1296 + 225} = 39 \)
❌ Mistake 2: Calculation errors in squaring
Writing \( 36^2 = 1206 \) instead of 1296
✓ Correct: \( 36 \times 36 = 1296 \) (check: \( 30^2 = 900, \, 6^2 = 36, \, 2 \times 30 \times 6 = 360 \))
❌ Mistake 3: Forgetting to take square root
Leaving answer as 1521 instead of 39
✓ Correct: Always complete the square root step: \( \sqrt{1521} = 39 \)
💡 Key Points to Remember
- Distance from origin: When one point is (0, 0), the formula becomes \( d = \sqrt{x^2 + y^2} \)
- Perfect squares: 1521 = 39² is a perfect square, making this problem simpler
- Real-world context: Coordinate geometry helps calculate actual distances between locations
- Units matter: Always check if the problem specifies a scale (e.g., 1 unit = 100 meters)
- Verification: You can check your answer by squaring it: 39² should equal 1521
- Pythagorean triple: (36, 15, 39) is related to the (12, 5, 13) triple multiplied by 3
📝 Practice Similar Problems
Strengthen your understanding:
