Beast Academy 5D — Pythagorean Paths Worksheet: Distance on the Coordinate Plane (Age 20, CCSS 8.G.B.8) — PNG Dot‑Grid √10, 5

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Instructions

Use the Pythagorean theorem (a² + b² = c²) to find the length of the diagonal segments on the coordinate grids below. The horizontal distance ("run") and vertical distance ("rise") between two points form the legs (a and b) of a right triangle. The length of the segment connecting the points is the hypotenuse (c). Express your answers in simplified radical form.

Part 1: Find the Length of Each Segment

Calculate the exact distance between the two marked points on each grid.

1. Length of segment AB: 
 A . . .
 . . . .
 . . . .
 . B . .
 . . . .
2. Length of segment CD: 
 . . . . D
 . . . . .
 . . . . .
 C . . . .
 . . . . .
3. Length of segment EF: 
 E . . . .
 . . . . .
 . . . . .
 . . . . .
 . . . F .
4. Length of segment GH: 
 . . . H .
 . . . . .
 . . . . .
 G . . . .
 . . . . .

Part 2: Calculate Total Path Length

The grid below shows a path traveling from point A to B, then to C, and finally to D. Calculate the length of each segment (AB, BC, and CD) and then find the total length of the entire path.

 . . . . . . . D
 . . . . . . . .
 . . . . . . . .
 . . . . C . . .
 . . . . . . . .
 A . . B . . . .
 . . . . . . . .
 . . . . . . . .

Length of AB: _______________
Length of BC: _______________
Length of CD: _______________

Total Path Length (A → B → C → D): _______________

Answer Key

Part 1: Find the Length of Each Segment

1. Length of segment AB: √10
The segment forms a right triangle with a rise of 3 units and a run of 1 unit.
a² + b² = c² => 1² + 3² = c² => 1 + 9 = c² => c² = 10 => c = √10.
2. Length of segment CD: √10
The segment forms a right triangle with a rise of 1 unit and a run of 3 units.
a² + b² = c² => 3² + 1² = c² => 9 + 1 = c² => c² = 10 => c = √10.
3. Length of segment EF: 5
The segment forms a right triangle with a rise of 4 units and a run of 3 units.
a² + b² = c² => 3² + 4² = c² => 9 + 16 = c² => c² = 25 => c = 5.
4. Length of segment GH: √10
The segment forms a right triangle with a rise of 3 units and a run of 1 unit.
a² + b² = c² => 1² + 3² = c² => 1 + 9 = c² => c² = 10 => c = √10.

Part 2: Calculate Total Path Length

Length of AB: √10
Rise = 1, Run = 3. => 3² + 1² = 10. Length = √10.
Length of BC: 5
Rise = 3, Run = 4. => 4² + 3² = 16 + 9 = 25. Length = 5.
Length of CD: √10
Rise = 3, Run = 1. => 1² + 3² = 1 + 9 = 10. Length = √10.
Total Path Length (A → B → C → D): 5 + 2√10
Total = (Length of AB) + (Length of BC) + (Length of CD) = √10 + 5 + √10 = 5 + 2√10.
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