Beast Academy 5 — Chapter 11 Worksheet: Square Roots & Pythagorean Theorem (Age 13, CCSS 8.G.B.6–8)

Instructions

Read each question carefully. The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle, c) is equal to the sum of the squares of the other two sides (the legs, a and b). The formula is: a² + b² = c².

Show your work where necessary. Don't forget to look for Pythagorean Triples as a potential shortcut!

Part 1: The Basics of the Theorem

1. A right triangle has legs measuring 7 cm and 24 cm. What is the length of the hypotenuse?


2. The hypotenuse of a right triangle is 25 inches long, and one of its legs is 15 inches long. How long is the other leg?


3. A 17-foot ladder is leaned against a vertical wall. The base of the ladder is 8 feet away from the base of the wall. How high up the wall does the ladder reach?


4. A rectangular park is 90 meters wide and 120 meters long. If you walk diagonally across the park from one corner to the opposite corner, how far have you walked?

Part 2: Pythagorean Triples

A Pythagorean Triple is a set of three positive integers (a, b, c) that perfectly satisfy the Pythagorean theorem. The most famous is (3, 4, 5). Any multiple of a triple is also a triple (e.g., 6, 8, 10).

5. From the list below, circle all the sets of numbers that are Pythagorean Triples.
   • (3, 4, 5)
   • (6, 7, 8)
   • (5, 12, 13)
   • (8, 15, 17)
   • (10, 24, 26)
   • (7, 20, 21)


6. A ship sails 30 miles east and then 40 miles north. How far is the ship from its starting point in a straight line? (Hint: Can you use a multiple of a common triple to solve this quickly?)


7. The two shorter sides of a right triangle are 20 and 21. What is the length of the longest side (the hypotenuse)? Is (20, 21, 29) a Pythagorean Triple? Explain why or why not.

Part 3: Pythagorean Paths

8. On a coordinate plane, find the direct, straight-line distance between Point A at (-2, 1) and Point B at (6, 7). Think of the distance as the hypotenuse of a right triangle formed by the horizontal and vertical distances.


9. An ant is on the bottom corner of a rectangular box that is 12 cm long, 9 cm wide, and 8 cm tall. The ant wants to crawl along the surface of the box to get to the opposite corner on the top of the box. What is the length of the shortest possible path the ant can take? (Hint: "Unfold" the box to make a flat surface and find the diagonal.)


10. An agent starts at point A(1, 2). She walks to point B(1, 6), and then to point C(7, 6).
    • a) What is the total distance the agent walked?
    • b) What is the straight-line distance from her starting point (A) to her final destination (C)?

Answer Key

1. 25 cm.
   a² + b² = c² → 7² + 24² = c² → 49 + 576 = 625 → c = √625 = 25.


2. 20 inches.
   a² + 15² = 25² → a² + 225 = 625 → a² = 400 → a = √400 = 20.


3. 15 feet.
   The ladder is the hypotenuse. h² + 8² = 17² → h² + 64 = 289 → h² = 225 → h = √225 = 15. This is an 8-15-17 Pythagorean triple.


4. 150 meters.
   The diagonal is the hypotenuse. 90² + 120² = d². This is a (3, 4, 5) triple scaled by 30. (3×30, 4×30, 5×30) = (90, 120, 150). The diagonal is 150 m.


5. The Pythagorean Triples are: (3, 4, 5), (5, 12, 13), (8, 15, 17), and (10, 24, 26) [which is 2 × (5, 12, 13)].


6. 50 miles.
   The path forms a right triangle with legs of 30 and 40. This is a (3, 4, 5) triple scaled by 10. (3×10, 4×10) means the hypotenuse is 5×10 = 50.


7. The hypotenuse is 29. Yes, it is a Pythagorean Triple.
   20² + 21² = c² → 400 + 441 = 841 → c = √841 = 29. It is a Pythagorean Triple because 20, 21, and 29 are all positive integers.


8. 10 units.
   The horizontal distance (leg a) is 6 - (-2) = 8. The vertical distance (leg b) is 7 - 1 = 6. The distance is the hypotenuse. 6² + 8² = c² → 36 + 64 = 100 → c = √100 = 10.


9. √433 cm (approximately 20.8 cm).
   To find the shortest path, unfold the box. The shortest path will be across the 12 cm length and 8 cm height, combined with the 9 cm width. Unfolding the front (12x8) and top (12x9) gives a flat rectangle of 12 cm by (8+9=17) cm. The diagonal is d² = 12² + 17² = 144 + 289 = 433. So, d = √433 cm. (Unfolding other ways gives longer paths, like √481 or √505).


10. • a) Total distance walked is 10 units.
      From A(1,2) to B(1,6) is 4 units. From B(1,6) to C(7,6) is 6 units. Total = 4 + 6 = 10.
    • b) Straight-line distance is 2√13 units (approx 7.21 units).
      This forms a right triangle with legs of 4 units (from A to B) and 6 units (from B to C). The distance AC is the hypotenuse. 4² + 6² = c² → 16 + 36 = 52 → c = √52. Simplified, √52 = √(4 × 13) = 2√13.