Pythagorean Paths: Enrichment Math Worksheet (Age 15) — ACARA v9 8.G.B.8 Distance Between Points (Years 8–12)

Instructions

To find the direct distance between two points on a coordinate grid, you can imagine a right-angled triangle where the distance you want to find is the hypotenuse (side 'c'). The horizontal and vertical lines connecting the points form the two other sides, or legs (sides 'a' and 'b').

You can then apply the Pythagorean Theorem: a² + b² = c².

1. Find the horizontal distance between the two points (the change in x-values). This is your 'a'.
2. Find the vertical distance between the two points (the change in y-values). This is your 'b'.
3. Substitute these values into the Pythagorean Theorem.
4. Solve for 'c' to find the distance. Remember to take the square root at the end!

Part 1: Guided Example

Let's find the distance between point A (-3, 2) and point B (5, 8).

• Step 1: Find the horizontal distance (a).
  Count the units between the x-coordinates -3 and 5. The distance is 5 - (-3) = 8 units. So, a = 8.
• Step 2: Find the vertical distance (b).
  Count the units between the y-coordinates 2 and 8. The distance is 8 - 2 = 6 units. So, b = 6.
• Step 3: Apply the Pythagorean Theorem.
  a² + b² = c²
  8² + 6² = c²
  64 + 36 = c²
  100 = c²
• Step 4: Solve for c.
  c = √100
  c = 10
  The distance between points A and B is 10 units.

Part 2: Practice Problems

For each pair of points below, calculate the distance. Show your work. Round your answers to two decimal places where necessary.

1. Find the distance between P (1, 1) and Q (6, 13).

2. Find the distance between R (-2, 5) and S (4, -3).

3. Find the distance between M (-7, -2) and N (-3, -5).

4. Find the distance between W (0, 5) and Z (8, -2).

Part 3: Application

Solve the following word problem. Be sure to show how you used the Pythagorean Theorem.

On a map of a nature preserve, the ranger station is located at (9, 4) and a scenic waterfall is located at (-3, -5). A straight trail is being built to connect the two locations.

If each unit on the map grid represents 50 meters, what will be the actual length of the new trail? Round your final answer to the nearest whole meter.

Answer Key

Part 2: Practice Problems

1. P (1, 1) and Q (6, 13)
   Horizontal distance (a) = 6 - 1 = 5
   Vertical distance (b) = 13 - 1 = 12
   a² + b² = c²
   5² + 12² = c²
   25 + 144 = c²
   169 = c²
   c = √169 = 13 units
2. R (-2, 5) and S (4, -3)
   Horizontal distance (a) = 4 - (-2) = 6
   Vertical distance (b) = 5 - (-3) = 8
   a² + b² = c²
   6² + 8² = c²
   36 + 64 = c²
   100 = c²
   c = √100 = 10 units
3. M (-7, -2) and N (-3, -5)
   Horizontal distance (a) = -3 - (-7) = 4
   Vertical distance (b) = -2 - (-5) = 3
   a² + b² = c²
   4² + 3² = c²
   16 + 9 = c²
   25 = c²
   c = √25 = 5 units
4. W (0, 5) and Z (8, -2)
   Horizontal distance (a) = 8 - 0 = 8
   Vertical distance (b) = 5 - (-2) = 7
   a² + b² = c²
   8² + 7² = c²
   64 + 49 = c²
   113 = c²
   c = √113 ≈ 10.63 units

Part 3: Application

Locations: Ranger Station (9, 4) and Waterfall (-3, -5)

• Step 1: Find the grid distance.
  Horizontal distance (a) = 9 - (-3) = 12
  Vertical distance (b) = 4 - (-5) = 9
  a² + b² = c²
  12² + 9² = c²
  144 + 81 = c²
  225 = c²
  c = √225 = 15 units
• Step 2: Convert grid distance to actual length.
  Each unit = 50 meters.
  Total length = 15 units * 50 meters/unit = 750 meters.

The actual length of the new trail will be 750 meters.