Pythagorean 2D

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Pythagorean 2D · Grade 8 · Worksheet 1

1. Charlotte is designing a rectangular garden for her backyard. The garden is 22 feet long and 17 feet wide. She wants to place a small decorative fountain at the center of the garden. What is the straight-line distance from one corner of the garden to the fountain? Round your answer to the nearest tenth of a foot. Answer: 13.9 Solution: The garden is 22 feet long and 17 feet wide. The center is at the midpoint of both dimensions. Half the length is 22 / 2 = 11 feet.
   Full step-by-step solution

   Step 1: The garden is 22 feet long and 17 feet wide. The center is at the midpoint of both dimensions. Step 2: Half the length is 22 / 2 = 11 feet. Step 3: Half the width is 17 / 2 = 8.5 feet. Step 4: The distance from a corner to the center forms the hypotenuse of a right triangle with legs 11 ft and 8.5 ft. Step 5: Apply the Pythagorean theorem: c^2 = 11^2 + 8.5^2. Step 6: 11^2 = 121 and 8.5^2 = 72.25. Step 7: Add the squares: 121 + 72.25 = 193.25. Step 8: Take the square root: c = sqrt(193.25). Step 9: sqrt(193.25) is approximately 13.901. Step 10: Round to the nearest tenth: 13.9. The answer is 13.9 feet.

2. Emma is helping her family build a wheelchair ramp for their front porch. The ramp will form a right triangle with the ground. The horizontal distance from the porch to the end of the ramp is 80 inches, and the vertical height from the ground to the porch is 60 inches. What is the length of the ramp (the diagonal side) in inches? Answer: 100 Solution: Identify the legs of the right triangle: horizontal distance = 80 inches, vertical height = 60 inches. The ramp is the hypotenuse.
   Full step-by-step solution

   Step 1: Identify the legs of the right triangle: horizontal distance = 80 inches, vertical height = 60 inches. The ramp is the hypotenuse. Step 2: Apply the Pythagorean theorem: a² + b² = c², where a = 80, b = 60. Step 3: Calculate 80² = 6400 Step 4: Calculate 60² = 3600 Step 5: Add the squares: 6400 + 3600 = 10000 Step 6: Find the square root: sqrt(10000) = 100 The ramp is 100 inches long.

3. Kaia is building a small ramp for her skateboard. The ramp forms a right triangle where the horizontal base (the distance from the start of the ramp to the point directly below the top) measures 9 inches, and the vertical height (the rise of the ramp) measures 12 inches. Kaia needs to cut a piece of plywood for the sloping surface of the ramp. How long, in inches, will the sloping surface be? Answer: 15 Solution: Identify the legs of the right triangle: base = 9 inches, height = 12 inches. The sloping surface is the hypotenuse (c). Use the Pythagorean theorem: a² + b² = c², where a = 9 and b = 12.
   Full step-by-step solution

   Step 1: Identify the legs of the right triangle: base = 9 inches, height = 12 inches. Step 2: The sloping surface is the hypotenuse (c). Use the Pythagorean theorem: a² + b² = c², where a = 9 and b = 12. Step 3: Calculate 9² = 81. Step 4: Calculate 12² = 144. Step 5: Add the squares: 81 + 144 = 225. Step 6: Find the square root: sqrt(225) = 15. The sloping surface of the ramp is 15 inches long.

4. √(37² - 12²) = ? Answer: 35 Solution: Write the Pythagorean theorem: a² + b² = c² We have c = 37 and b = 12, so we need to find a: a² + 12² = 37² Calculate the squares: a² + 144 = 1369 Subtract 144 from both sides: a² = 1369 - 144 Simplify: a² = 1225 Take the square root: a = √1225 √1225 = 35 The answer is 35.
   Full step-by-step solution

   Step 1: Write the Pythagorean theorem: a² + b² = c² Step 2: We have c = 37 and b = 12, so we need to find a: a² + 12² = 37² Step 3: Calculate the squares: a² + 144 = 1369 Step 4: Subtract 144 from both sides: a² = 1369 - 144 Step 5: Simplify: a² = 1225 Step 6: Take the square root: a = √1225 Step 7: √1225 = 35 The answer is 35.

5. Liam is building a ramp for his skateboard. The ramp forms a right triangle with the ground. The vertical height of the ramp is 7 feet, and the horizontal distance from the base of the ramp to the point directly below the top is 9 feet. What is the length of the ramp (the hypotenuse) in feet? Round your answer to the nearest tenth of a foot. Answer: 11.4 Solution: Identify the legs of the right triangle. The vertical height is 7 ft, and the horizontal distance is 9 ft. The ramp is the hypotenuse.
   Full step-by-step solution

   Step 1: Identify the legs of the right triangle. The vertical height is 7 ft, and the horizontal distance is 9 ft. The ramp is the hypotenuse. Step 2: Apply the Pythagorean theorem: a^2 + b^2 = c^2, where a = 7, b = 9, and c is the ramp length. Step 3: Calculate 7^2 = 49 and 9^2 = 81. Step 4: Add the squares: 49 + 81 = 130. Step 5: Take the square root: c = sqrt(130). Step 6: Since 130 is not a perfect square, simplify: sqrt(130) ≈ 11.4018. Step 7: Round to the nearest tenth: 11.4. The answer is 11.4 feet.