Pythagorean 3D

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Pythagorean 3D · Grade 8 · Worksheet 1

1. 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.

2. A rectangular prism has dimensions 6 cm × 8 cm × 24 cm. Find the length of the space diagonal from one vertex to the opposite vertex. Answer: 26 cm Solution: We are given a rectangular prism with dimensions 6 cm, 8 cm, and 24 cm. We want the length of the space diagonal from one vertex to the opposite vertex.
   Full step-by-step solution

   We are given a rectangular prism with dimensions 6 cm, 8 cm, and 24 cm. We want the length of the space diagonal from one vertex to the opposite vertex. Step 1: Understand the space diagonal formula. For a rectangular prism with length \( l \), width \( w \), and height \( h \), the space diagonal \( d \) is given by: \[ d = \sqrt{l^2 + w^2 + h^2} \] Step 2: Identify \( l, w, h \). Here, \( l = 6 \) cm, \( w = 8 \) cm, \( h = 24 \) cm. Step 3: Compute the squares. \[ l^2 = 6^2 = 36 \] \[ w^2 = 8^2 = 64 \] \[ h^2 = 24^2 = 576 \] Step 4: Add the squares. \[ l^2 + w^2 + h^2 = 36 + 64 + 576 \] First, \( 36 + 64 = 100 \) Then \( 100 + 576 = 676 \) Step 5: Take the square root. \[ d = \sqrt{676} \] We know \( 26^2 = 676 \), so \( \sqrt{676} = 26 \). Step 6: State the final answer. The length of the space diagonal is 26 cm. Final answer: 26 cm

3. √(9² + 12² + 20²) = ? Answer: 25 Solution: Square each dimension: 9² = 81, 12² = 144, 20² = 400 Add the squares: 81 + 144 + 400 = 625 Take the square root: √625 = 25 The answer is 25.
   Full step-by-step solution

   Step 1: Square each dimension: 9² = 81, 12² = 144, 20² = 400 Step 2: Add the squares: 81 + 144 + 400 = 625 Step 3: Take the square root: √625 = 25 The answer is 25.

4. √(18² + 24² + 80²) = ? Answer: 86 Solution: Square each dimension: 18² = 324, 24² = 576, 80² = 6400 Add the squares: 324 + 576 + 6400 = 7300 Take the square root: √7300 = 86 The answer is 86.
   Full step-by-step solution

   Step 1: Square each dimension: 18² = 324, 24² = 576, 80² = 6400 Step 2: Add the squares: 324 + 576 + 6400 = 7300 Step 3: Take the square root: √7300 = 86 The answer is 86.

5. √(15² + 20² + 36²) = ? Answer: 45 Solution: Calculate the diagonal of the base rectangle using the Pythagorean theorem: √(15² + 20²) = √(225 + 400) = √625 = 25 Use this diagonal with the height to find the space diagonal: √(25² + 36²) = √(625 + 1296) = √1921 = 45 Alternatively, use the 3D Pythagorean theorem directly: √(15² + 20² + 36²) =…
   Full step-by-step solution

   Step 1: Calculate the diagonal of the base rectangle using the Pythagorean theorem: √(15² + 20²) = √(225 + 400) = √625 = 25 Step 2: Use this diagonal with the height to find the space diagonal: √(25² + 36²) = √(625 + 1296) = √1921 = 45 Step 3: Alternatively, use the 3D Pythagorean theorem directly: √(15² + 20² + 36²) = √(225 + 400 + 1296) = √1921 = 45 The answer is 45.