Pythagorean 2D

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Answer Key & Explanations

Pythagorean 2D · Grade 8 · Worksheet 3

1. A right triangle has legs of length 7 cm and 24 cm. What is the length of the hypotenuse? Answer: 25 cm Solution: We are given a right triangle with legs of length 7 cm and 24 cm. We need to find the length of the hypotenuse. Recall the Pythagorean theorem.
   Full step-by-step solution

   We are given a right triangle with legs of length 7 cm and 24 cm. We need to find the length of the hypotenuse. Step 1: Recall the Pythagorean theorem. For a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. If the legs are a and b, and the hypotenuse is c, then: a^2 + b^2 = c^2 Step 2: Substitute the given lengths. Here, a = 7 cm, b = 24 cm. So: 7^2 + 24^2 = c^2 Step 3: Calculate the squares. 7^2 = 49 24^2 = 576 So: 49 + 576 = c^2 Step 4: Add the results. 49 + 576 = 625 Therefore: c^2 = 625 Step 5: Find c by taking the square root. c = square root of 625 Since 25 * 25 = 625, we have: c = 25 Step 6: State the final answer. The length of the hypotenuse is 25 cm.

2. Noah is building a rectangular garden that measures 16 meters by 11 meters. He wants to install a diagonal sprinkler line from one corner to the opposite corner. What is the length of the diagonal sprinkler line? Round your answer to the nearest whole meter. Answer: 19 Solution: The garden is a rectangle with length = 16 m and width = 11 m. The diagonal is the hypotenuse of a right triangle where the legs are the length and width. Calculate 16^2 = 256 and 11^2 = 121.
   Full step-by-step solution

   Step 1: The garden is a rectangle with length = 16 m and width = 11 m. The diagonal is the hypotenuse of a right triangle where the legs are the length and width. Step 2: Apply the Pythagorean theorem: a^2 + b^2 = c^2, where a = 16, b = 11, and c is the diagonal. Step 3: Calculate 16^2 = 256 and 11^2 = 121. Step 4: Add the squares: 256 + 121 = 377. Step 5: Take the square root: c = sqrt(377). Step 6: sqrt(377) is approximately 19.416. Step 7: Round to the nearest whole meter: 19. The answer is 19 meters.

3. Liam is building a triangular support brace for his bookshelf. The brace will be a right triangle where the horizontal piece measures 48 cm and the vertical piece measures 64 cm. What is the length of the diagonal support beam that Liam needs to cut? Answer: 80 cm Solution: We are told that the brace is a right triangle. The horizontal piece is 48 cm, and the vertical piece is 64 cm. The diagonal support beam is the hypotenuse of this right triangle.
   Full step-by-step solution

   We are told that the brace is a right triangle. The horizontal piece is 48 cm, and the vertical piece is 64 cm. The diagonal support beam is the hypotenuse of this right triangle. Step 1: Recall the Pythagorean theorem. For a right triangle with legs a and b and hypotenuse c: a^2 + b^2 = c^2. Step 2: Assign values. Let a = 48 cm, b = 64 cm, and c = the diagonal we need to find. Step 3: Apply the theorem. c^2 = 48^2 + 64^2. Step 4: Calculate the squares. 48^2 = 48 × 48 = 2304 64^2 = 64 × 64 = 4096 Step 5: Add them. 2304 + 4096 = 6400 Step 6: Find c by taking the square root. c = sqrt(6400) c = 80 Step 7: Conclusion. The length of the diagonal support beam is 80 cm.

4. A rectangular park has a length of 24 meters and a width of 10 meters. A diagonal path is planned from one corner of the park to the opposite corner. Additionally, a circular fountain is to be placed in the center of the park such that its circumference touches the midpoint of each side of the rectangle. What is the straight-line distance from the center of the fountain to one of the corners of the park? Round your answer to the nearest tenth of a meter. Answer: 13.0 Solution: Find the center of the rectangle. The center is at the midpoint of both the length and the width. Half the length is 24 / 2 = 12 meters.
   Full step-by-step solution

   Step 1: Find the center of the rectangle. The center is at the midpoint of both the length and the width. Step 2: Half the length is 24 / 2 = 12 meters. Step 3: Half the width is 10 / 2 = 5 meters. Step 4: The distance from the center to a corner forms the hypotenuse of a right triangle with legs of 12 m and 5 m. Step 5: Apply the Pythagorean Theorem: distance^2 = 12^2 + 5^2. Step 6: Calculate 12^2 = 144 and 5^2 = 25. Step 7: Add the squares: 144 + 25 = 169. Step 8: Find the square root: sqrt(169) = 13. Step 9: The problem asks for the answer rounded to the nearest tenth, so 13.0. The final answer is 13.0 meters.

5. √(20² + 21²) = ? Answer: 29 Solution: Square the first number: 20² = 400 Square the second number: 21² = 441 Add the squares: 400 + 441 = 841 Find the square root: √841 = 29 The answer is 29.
   Full step-by-step solution

   Step 1: Square the first number: 20² = 400 Step 2: Square the second number: 21² = 441 Step 3: Add the squares: 400 + 441 = 841 Step 4: Find the square root: √841 = 29 The answer is 29.