Pythagorean 3D

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Answer Key & Explanations

Pythagorean 3D · Grade 8 · Worksheet 2

1. Liam is building a triangular support brace for his bookshelf. The brace needs to fit diagonally across the back of the shelf, which is 24 inches wide and 18 inches tall. What is the exact length of the diagonal brace Liam needs to cut? Answer: 30 inches Solution: We are given a bookshelf that is 24 inches wide and 18 inches tall, and we need to find the diagonal length of the triangular brace. Recognize the shape.
   Full step-by-step solution

   We are given a bookshelf that is 24 inches wide and 18 inches tall, and we need to find the diagonal length of the triangular brace. Step 1: Recognize the shape. The shelf forms a rectangle, and the diagonal divides it into two right triangles. The width and height are the two legs of the right triangle, and the diagonal is the hypotenuse. Step 2: Recall the Pythagorean theorem. For a right triangle with legs a and b and hypotenuse c: a^2 + b^2 = c^2 Step 3: Assign values. Let a = width = 24 inches Let b = height = 18 inches Let c = diagonal length (unknown) Step 4: Apply the theorem. c^2 = a^2 + b^2 c^2 = 24^2 + 18^2 Step 5: Calculate squares. 24^2 = 576 18^2 = 324 Step 6: Add the squares. c^2 = 576 + 324 c^2 = 900 Step 7: Take the square root. c = square root of 900 c = 30 Step 8: State the answer. The exact length of the diagonal brace is 30 inches.

2. A rectangular prism has dimensions 11 cm × 16 cm × 26 cm. Find the length of the space diagonal from one vertex to the opposite vertex. Answer: √(11² + 16² + 26²) = √(121 + 256 + 676) = √1053 = 9√13 ≈ 32.45 cm Solution: Identify the dimensions: length = 11 cm, width = 16 cm, height = 26 cm. Find the diagonal of the base (the rectangle formed by length and width).
   Full step-by-step solution

   Step 1: Identify the dimensions: length = 11 cm, width = 16 cm, height = 26 cm. Step 2: Find the diagonal of the base (the rectangle formed by length and width). Use the Pythagorean theorem: base diagonal = √(11² + 16²) = √(121 + 256) = √377. Step 3: Now, the space diagonal is the hypotenuse of a right triangle with legs equal to the base diagonal and the height. Apply the Pythagorean theorem again: space diagonal = √((√377)² + 26²) = √(377 + 676) = √1053. Step 4: Simplify √1053. Factor 1053: 1053 = 9 × 117 = 9 × 9 × 13 = 81 × 13. So √1053 = √(81 × 13) = 9√13. Step 5: The exact length is 9√13 cm. As a decimal, 9√13 ≈ 9 × 3.6055 = 32.45 cm. The answer is 9√13 cm (approximately 32.45 cm).

3. Mere is building a display case for a museum artifact. The case is a rectangular prism with a length of 8 cm, a width of 6 cm, and a height of 4 cm. She wants to place a laser pointer at the bottom front left corner so that the beam travels in a straight line to the top back right corner of the case. What is the length, in centimeters, of the laser beam's path through the case? Answer: sqrt(116) or 10.8 Solution: Find the diagonal of the base (length and width). Use the Pythagorean theorem: d1^2 = 8^2 + 6^2 = 64 + 36 = 100, so d1 = sqrt(100) = 10 cm.
   Full step-by-step solution

   Step 1: Find the diagonal of the base (length and width). Use the Pythagorean theorem: d1^2 = 8^2 + 6^2 = 64 + 36 = 100, so d1 = sqrt(100) = 10 cm. Step 2: Now consider a vertical right triangle where one leg is the base diagonal (10 cm) and the other leg is the height (4 cm). The space diagonal (the laser beam) is the hypotenuse. Step 3: Apply the Pythagorean theorem again: d2^2 = 10^2 + 4^2 = 100 + 16 = 116. Step 4: Take the square root: d2 = sqrt(116) = sqrt(4*29) = 2*sqrt(29). Approximating: sqrt(116) ≈ 10.770... Rounded to the nearest tenth, it is 10.8 cm. The length of the laser beam's path is sqrt(116) cm, or approximately 10.8 cm.

4. Kaia is constructing a large rectangular prism frame for a school art installation. The interior dimensions of the frame are 11 cm long, 9 cm wide, and 7 cm high. She needs to run a single straight string of LED lights from the bottom front left corner to the top back right corner of the frame to create a glowing diagonal effect. What is the exact length, in centimeters, of the longest straight string of lights that can fit inside the frame? Simplify the square root if possible. Answer: sqrt(251) Solution: Identify the dimensions of the frame: length = 11 cm, width = 9 cm, height = 7 cm.
   Full step-by-step solution

   Step 1: Identify the dimensions of the frame: length = 11 cm, width = 9 cm, height = 7 cm. Step 2: Find the diagonal of the base (the floor) using the Pythagorean theorem on the base rectangle: base diagonal^2 = length^2 + width^2 = 11^2 + 9^2 = 121 + 81 = 202. So the base diagonal = sqrt(202). Step 3: Now consider the right triangle formed by the base diagonal (sqrt(202)), the height (7 cm), and the space diagonal (the string of lights). The space diagonal is the hypotenuse. Step 4: Apply the Pythagorean theorem again: space diagonal^2 = (sqrt(202))^2 + 7^2 = 202 + 49 = 251. Step 5: Take the square root: space diagonal = sqrt(251). Since 251 is a prime number (it has no perfect square factors other than 1), it cannot be simplified further. Step 6: The exact length of the longest straight string of lights is sqrt(251) cm.