Pythagorean Paths — Year 8 Maths (Age 13): Distance Between Points + ACARA v9 Rubrics

Instructions

Welcome, intrepid mathematician! Your mission is to navigate a coordinate plane and find the shortest distance between two points. This direct path is the hypotenuse of a right triangle.

To find this distance, you will use the Pythagorean Theorem: a² + b² = c².

• Imagine a right triangle connecting your two points.
• The horizontal distance (the change in x-values) is one leg, 'a'.
• The vertical distance (the change in y-values) is the other leg, 'b'.
• The direct path you want to find is the hypotenuse, 'c'.

For answers that are not whole numbers, you may leave them as a simplified square root or round to the nearest tenth.

Part 1: Finding Your Footing

Calculate the distance between the following pairs of points. Show your work.

1. Point A = (2, 1) and Point B = (5, 5)


2. Point C = (-3, 2) and Point D = (9, 7)


3. Point E = (1, 9) and Point F = (7, 1)


4. Point G = (-4, -2) and Point H = (1, -8)

Part 2: The Explorer's Journey

An explorer is mapping a newly discovered island on a coordinate grid. Each unit on the grid represents one kilometer. She must travel in straight lines from one landmark to the next.

• She begins at Base Camp located at (0, 0).
• Her first stop is the Crystal Cave at (12, 5).
• From there, she travels to the Hidden Waterfall at (15, 9).
• Her final destination for the day is the Summit at (15, 2).

Calculate the length of each leg of her journey and the total distance traveled.

1. What is the distance from Base Camp to the Crystal Cave?


2. What is the distance from the Crystal Cave to the Hidden Waterfall?


3. What is the distance from the Hidden Waterfall to the Summit? (Hint: Look closely at the coordinates!)


4. What is the total distance the explorer traveled?

Part 3: The Race to the Park

Two friends, Mia and Leo, agree to meet at the park. The park is located at point P = (2, 1).

• Mia is at her house at point M = (-4, -7).
• Leo is at the library at point L = (10, 6).

They both travel in a straight line to the park. Who has the shorter path?

1. Calculate the distance from Mia's house (M) to the park (P).


2. Calculate the distance from the library (L) to the park (P).


3. Who has the shorter path to the park?

Answer Key

Part 1: Finding Your Footing

1. Distance = 5 units.
   Horizontal distance 'a' = 5 - 2 = 3. Vertical distance 'b' = 5 - 1 = 4.
   3² + 4² = c² → 9 + 16 = 25 → c = √25 = 5.
2. Distance = 13 units.
   Horizontal distance 'a' = 9 - (-3) = 12. Vertical distance 'b' = 7 - 2 = 5.
   12² + 5² = c² → 144 + 25 = 169 → c = √169 = 13.
3. Distance = 10 units.
   Horizontal distance 'a' = 7 - 1 = 6. Vertical distance 'b' = 9 - 1 = 8.
   6² + 8² = c² → 36 + 64 = 100 → c = √100 = 10.
4. Distance = √61 units ≈ 7.8 units.
   Horizontal distance 'a' = 1 - (-4) = 5. Vertical distance 'b' = -2 - (-8) = 6.
   5² + 6² = c² → 25 + 36 = 61 → c = √61.

Part 2: The Explorer's Journey

1. 13 km.
   a = 12 - 0 = 12. b = 5 - 0 = 5.
   12² + 5² = c² → 144 + 25 = 169 → c = 13.
2. 5 km.
   a = 15 - 12 = 3. b = 9 - 5 = 4.
   3² + 4² = c² → 9 + 16 = 25 → c = 5.
3. 7 km.
   The x-coordinates (15) are the same, so this is a straight vertical line. The distance is the difference in y-coordinates: 9 - 2 = 7. (No theorem needed!)
4. Total distance = 25 km.
   13 km + 5 km + 7 km = 25 km.

Part 3: The Race to the Park

1. Mia's distance is 10 units.
   Horizontal distance 'a' = 2 - (-4) = 6. Vertical distance 'b' = 1 - (-7) = 8.
   6² + 8² = c² → 36 + 64 = 100 → c = 10.
2. Leo's distance is √89 ≈ 9.4 units.
   Horizontal distance 'a' = 10 - 2 = 8. Vertical distance 'b' = 6 - 1 = 5.
   8² + 5² = c² → 64 + 25 = 89 → c = √89.
3. Leo has the shorter path. (9.4 is less than 10).