Pythagorean 3D

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Answer Key & Explanations

Pythagorean 3D · Grade 8 · Worksheet 3

1. A rectangular pyramid has a base measuring 12 cm by 16 cm and a vertical height of 15 cm from the center of the base to the apex. What is the length of the slant edge from the apex to one of the base's corners? Round your answer to the nearest tenth of a centimeter. Answer: 18.0 Solution: Find the diagonal of the rectangular base. The base is 12 cm by 16 cm. Use the Pythagorean theorem: base_diagonal = sqrt(12^2 + 16^2) = sqrt(144 + 256) = sqrt(400) = 20 cm.
   Full step-by-step solution

   Step 1: Find the diagonal of the rectangular base. The base is 12 cm by 16 cm. Use the Pythagorean theorem: base_diagonal = sqrt(12^2 + 16^2) = sqrt(144 + 256) = sqrt(400) = 20 cm. Step 2: The distance from the center of the base to a corner is half the diagonal: 20 / 2 = 10 cm. Step 3: The vertical height from the center of the base to the apex is 15 cm. This forms a right triangle with the slant edge as the hypotenuse. Step 4: Apply the Pythagorean theorem again: slant_edge = sqrt(10^2 + 15^2) = sqrt(100 + 225) = sqrt(325) ≈ 18.02776 cm. Step 5: Round to the nearest tenth: 18.0 cm. The length of the slant edge is 18.0 cm.

2. Kaia has a rectangular storage shed. The floor is 7 feet by 9 feet, and the vertical height from the floor to the ceiling is 11 feet. A support cable runs in a straight line from the bottom front left corner of the floor to the top back right corner of the ceiling. What is the length of this cable? Round your answer to the nearest tenth of a foot. Answer: 15.8 Solution: Identify the dimensions: length = 9 ft, width = 7 ft, height = 11 ft. Find the diagonal of the floor. Use the Pythagorean theorem: floor diagonal = sqrt(9^2 + 7^2) = sqrt(81 + 49) = sqrt(130) ft.
   Full step-by-step solution

   Step 1: Identify the dimensions: length = 9 ft, width = 7 ft, height = 11 ft. Step 2: Find the diagonal of the floor. Use the Pythagorean theorem: floor diagonal = sqrt(9^2 + 7^2) = sqrt(81 + 49) = sqrt(130) ft. Step 3: The cable is the space diagonal. Form a right triangle with legs: floor diagonal (sqrt(130) ft) and height (11 ft). The cable is the hypotenuse. Step 4: Apply the Pythagorean theorem again: cable length = sqrt((sqrt(130))^2 + 11^2) = sqrt(130 + 121) = sqrt(251). Step 5: Calculate sqrt(251). 15^2 = 225, 16^2 = 256, so sqrt(251) is between 15 and 16. More precisely: 15.8^2 = 249.64, 15.9^2 = 252.81, so sqrt(251) ≈ 15.843. Step 6: Round to the nearest tenth: 15.8 ft. The length of the cable is 15.8 feet.

3. Emma is designing a large art installation in the shape of a rectangular prism. The base of the prism is 15 feet long and 20 feet wide. The height of the prism is 25 feet. A support cable runs in a straight line from the bottom front left corner of the prism to the top back right corner. What is the length of this support cable, to the nearest foot? Answer: 35 Solution: The base is a rectangle 15 ft by 20 ft. Find the base diagonal: sqrt(15^2 + 20^2) = sqrt(225 + 400) = sqrt(625) = 25 ft. Step 2: The cable is the space diagonal.
   Full step-by-step solution

   Step 1: The base is a rectangle 15 ft by 20 ft. Find the base diagonal: sqrt(15^2 + 20^2) = sqrt(225 + 400) = sqrt(625) = 25 ft. Step 2: The cable is the space diagonal. Form a right triangle with legs: base diagonal (25 ft) and height (25 ft). Step 3: Apply Pythagorean theorem: cable length = sqrt(25^2 + 25^2) = sqrt(625 + 625) = sqrt(1250). Step 4: sqrt(1250) = sqrt(625 * 2) = 25 * sqrt(2) ≈ 25 * 1.414 = 35.35 ft. Step 5: Round to the nearest foot: 35 ft. The cable length is 35 feet.

4. Liam is building a triangular support brace for a bookshelf. The brace forms a right triangle with the shelf and the wall. The horizontal distance along the shelf is 24 cm, and the vertical distance up the wall is 32 cm. What is the length of the diagonal brace? Round your answer to the nearest centimeter. Answer: 40 Solution: - The horizontal leg (along the shelf) = 24 cm - The vertical leg (up the wall) = 32 cm - The diagonal brace is the hypotenuse. Recall the Pythagorean theorem.
   Full step-by-step solution

   We are given a right triangle where: - The horizontal leg (along the shelf) = 24 cm - The vertical leg (up the wall) = 32 cm - The diagonal brace is the hypotenuse. 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: Substitute the given lengths. Here, a = 24, b = 32, c = unknown brace length. So: 24^2 + 32^2 = c^2 Step 3: Calculate the squares. 24^2 = 576 32^2 = 1024 Step 4: Add them. 576 + 1024 = 1600 Step 5: Now we have: c^2 = 1600 Step 6: Take the square root of both sides. c = sqrt(1600) c = 40 Step 7: Round to the nearest centimeter. 40 is already a whole number. Final answer: The length of the diagonal brace is 40 cm.