Distance Formula Quiz Document Coordinate Cartography
Distance Formula Quick Check
Explorer:
Date:
Mission: Find the exact distance between the given points. Show all steps and simplify all radical answers (e.g., \(\sqrt{18} = 3\sqrt{2}\)).
1
Distance between \(A(2, 3)\) and \(B(5, 9)\).
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2
Distance between \(C(-1, 4)\) and \(D(3, -2)\).
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3
Landmark Route: Bakery to Library
The Bakery is at \((1, 2)\) and the Library is at \((7, 10)\). If 1 unit = 1 mile, what is the straight-line distance?
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4
Scenic Route: Park to School
The Entrance is at \((-2, -5)\) and the School is at \((4, 1)\). Find the exact distance.
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5
The Treasure Challenge
A treasure is \(5\) units from camp \((1, 1)\) at \((4, y)\). Find one possible value of \(y\).
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Cartography Series
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Distance Formula Quiz Key Document Coordinate Cartography
Official Answer Key
Scoring Key
1
Points \(A(2, 3), B(5, 9)\)
\(d = \sqrt{(5-2)^2 + (9-3)^2}\)
\(d = \sqrt{3^2 + 6^2}\)
\(d = \sqrt{45} = 3\sqrt{5}\)
2
Points \(C(-1, 4), D(3, -2)\)
\(d = \sqrt{(3-(-1))^2 + (-2-4)^2}\)
\(d = \sqrt{4^2 + (-6)^2}\)
\(d = \sqrt{52} = 2\sqrt{13}\)
3
Bakery \((1, 2)\) to Library \((7, 10)\)
\(d = \sqrt{(7-1)^2 + (10-2)^2}\)
\(d = \sqrt{36 + 64} = 10 \text{ miles}\)
Check for "miles" units.
4
Park \((-2, -5)\) to School \((4, 1)\)
\(d = \sqrt{(4 - (-2))^2 + (1 - (-5))^2}\)
\(d = \sqrt{6^2 + 6^2} = \sqrt{72}\)
\(d = 6\sqrt{2}\)
5
Treasure: Camp \((1, 1)\), Treasure \((4, y)\), Dist \(5\)
\(25 = (4-1)^2 + (y-1)^2 \rightarrow 25 = 9 + (y-1)^2\)
\(16 = (y-1)^2 \rightarrow y-1 = \pm 4 \rightarrow y \in \{5, -3\}\)
Scoring Guidelines
Point Breakdown
1.0 pt: Correct substitution
0.5 pt: Calculation accuracy
0.5 pt: Simplest radical form
Common Mistakes
Sign errors in subtractions
Square root of negatives errors
Unsimplified radicals (e.g., \(\sqrt{45}\))
Distance Formula Intro Slides Geometry Expedition
Coordinate
Cartography
Navigating the grid with the Distance Formula.
The Distance Formula
Derived from the Pythagorean Theorem, this formula calculates the precise distance between two points.
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Cartographer's Checklist
Label coordinates (x₁, y₁) and (x₂, y₂) .
Subtract carefully. Watch for double negatives!
Simplify the radical for the final answer.
Exact Measurements
Cartographers prefer Simplest Radical Form over decimal approximations.
1. Factor
Find the largest perfect square factor.
\(\sqrt{50}\)
2. Split
Rewrite as a product of two radicals.
\(\sqrt{25} \cdot \sqrt{2}\)
3. Simplify
Calculate the root of the perfect square.
\(5\sqrt{2}\)
Example: The Coastal Lighthouse
A lighthouse is at (-3, 2). A boat is at (5, -2).
What is the exact distance between these two points?
Solution Path
1
\(d = \sqrt{(5 - (-3))^2 + (-2 - 2)^2}\)
2
\(d = \sqrt{8^2 + (-4)^2}\)
3
\(d = \sqrt{64 + 16} = \sqrt{80}\)
4
\(d = \sqrt{16 \cdot 5}\)
ANS
\(4\sqrt{5}\)
Expedition Ready?
Prepare your compass and your calculations. The mission includes pure geometry and map applications.
5
Questions
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Exact Roots
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