Pythagorean 2D

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Pythagorean 2D · Grade 8 · Worksheet 2

1. A rectangular garden is 12 meters long and 5 meters wide. A diagonal path is built directly from one corner to the opposite corner. What is the length of this diagonal path? Round your answer to the nearest tenth of a meter. Answer: 13.0 Solution: We are given a rectangular garden that is 12 meters long and 5 meters wide. A diagonal path runs from one corner to the opposite corner. We need to find the length of this diagonal.
   Full step-by-step solution

   We are given a rectangular garden that is 12 meters long and 5 meters wide. A diagonal path runs from one corner to the opposite corner. We need to find the length of this diagonal. Step 1: Understand the problem. The diagonal of a rectangle forms a right triangle with the length and width as the two legs. We can use the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the legs, and c is the diagonal. Step 2: Assign values. Length = 12 m, width = 5 m. So: a = 12, b = 5, c = diagonal. Step 3: Apply the Pythagorean theorem. c^2 = 12^2 + 5^2 c^2 = 144 + 25 c^2 = 169 Step 4: Solve for c. c = square root of 169 c = 13 Step 5: Round to the nearest tenth. 13.0 is already to the nearest tenth. Step 6: Final answer. The length of the diagonal path is 13.0 meters.

2. Emma is designing a kite for a school project. The kite has a rectangular frame with a length of 24 inches and a width of 10 inches. She wants to add a diagonal cross-brace for stability. What is the length, in inches, of this diagonal brace? Round your answer to the nearest tenth of an inch. Answer: 26.0 Solution: The diagonal of a rectangle forms a right triangle with the length and width as the two legs. Substitute the known values: 24^2 + 10^2 = c^2 Calculate the squares: 576 + 100 = c^2 Add the results: 676 = c^2 Take the square root of both sides: c = sqrt(676) Calculate the square root: c = 26 The…
   Full step-by-step solution

   Step 1: The diagonal of a rectangle forms a right triangle with the length and width as the two legs. Step 2: Apply the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the sides of the rectangle, and c is the diagonal. Step 3: Substitute the known values: 24^2 + 10^2 = c^2 Step 4: Calculate the squares: 576 + 100 = c^2 Step 5: Add the results: 676 = c^2 Step 6: Take the square root of both sides: c = sqrt(676) Step 7: Calculate the square root: c = 26 Step 8: The problem asks for the answer rounded to the nearest tenth, so 26.0 The length of the diagonal brace is 26.0 inches.

3. Mere is constructing a rectangular garden for a community project. The garden has a length of 18 meters and a width of 24 meters. She wants to install a diagonal irrigation pipe from one corner to the opposite corner. What is the length of the diagonal pipe, in meters? Answer: 30 Solution: The garden forms a right triangle with legs of 18 meters and 24 meters. Let a = 18, b = 24, and c be the diagonal (hypotenuse).
   Full step-by-step solution

   Step 1: The garden forms a right triangle with legs of 18 meters and 24 meters. Step 2: Let a = 18, b = 24, and c be the diagonal (hypotenuse). Step 3: Apply the Pythagorean theorem: a² + b² = c² Step 4: Substitute: 18² + 24² = c² Step 5: Calculate: 324 + 576 = c² Step 6: Add: 900 = c² Step 7: Take the square root: c = sqrt(900) = 30 The diagonal pipe is 30 meters long.

4. √(20² - 12²) = ? Answer: 16 Solution: Write the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Let the unknown leg be x.
   Full step-by-step solution

   Step 1: Write the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Step 2: In this problem, we have c = 20 and one leg = 12. Let the unknown leg be x. Step 3: Set up the equation: x² + 12² = 20² Step 4: Calculate the squares: x² + 144 = 400 Step 5: Subtract 144 from both sides: x² = 400 - 144 Step 6: Simplify: x² = 256 Step 7: Take the square root of both sides: x = √256 Step 8: Calculate: x = 16 The answer is 16.