Pythagorean Theorem

lessonbunny.com

Answer Key & Explanations

Pythagorean Theorem · Grade 8 · Worksheet 1

1. Lena is designing a triangular garden plot with sides measuring 6 meters and 8 meters. She wants to install a diagonal stone path across the plot. What is the length of this diagonal path? Answer: 10 Solution: Lena has a triangular garden plot with sides 6 m and 8 m, and she wants a diagonal stone path across it. If the triangle is right-angled, the diagonal across the two shorter sides is the hypotenuse.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** Lena has a triangular garden plot with sides 6 m and 8 m, and she wants a diagonal stone path across it. If the triangle is right-angled, the diagonal across the two shorter sides is the hypotenuse. --- **Step 2: Identify the right triangle** Assume the two given sides (6 m and 8 m) are the legs of a right triangle, and the diagonal is the hypotenuse. This fits the common 3-4-5 triangle scaled up: 6-8-10. --- **Step 3: Apply the Pythagorean theorem** The Pythagorean theorem says: a^2 + b^2 = c^2 where a and b are the legs, c is the hypotenuse. Here: a = 6, b = 8, c = diagonal path length. --- **Step 4: Calculate step-by-step** First, square each leg: 6^2 = 36 8^2 = 64 Add them: 36 + 64 = 100 Now, c^2 = 100, so c = square root of 100. c = 10 --- **Step 5: Conclusion** The length of the diagonal stone path is 10 meters. --- **Final answer:** 10

2. √(11² + 60²) = ? Answer: 61 Solution: Calculate 11 squared: 11² = 121 Calculate 60 squared: 60² = 3600 Add the two squared values: 121 + 3600 = 3721 Take the square root of the sum: √3721 = 61 The answer is 61.
   Full step-by-step solution

   Step 1: Calculate 11 squared: 11² = 121 Step 2: Calculate 60 squared: 60² = 3600 Step 3: Add the two squared values: 121 + 3600 = 3721 Step 4: Take the square root of the sum: √3721 = 61 The answer is 61.

3. √(9² + 40²) = ? Answer: 41 Solution: Calculate 9 squared: 9² = 81 Calculate 40 squared: 40² = 1600 Add the two squared values: 81 + 1600 = 1681 Take the square root of the sum: √1681 = 41 The answer is 41.
   Full step-by-step solution

   Step 1: Calculate 9 squared: 9² = 81 Step 2: Calculate 40 squared: 40² = 1600 Step 3: Add the two squared values: 81 + 1600 = 1681 Step 4: Take the square root of the sum: √1681 = 41 The answer is 41.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A square is constructed on each side of the triangle, with each square's side being one of the triangle's sides. What is the total area of the three squares? Answer: 200 Solution: A = (0,0) B = (6,0) C = (6,8) Side AB: from (0,0) to (6,0) This is horizontal, length = 6 - 0 = 6. Side BC: from (6,0) to (6,8) This is vertical, length = 8 - 0 = 8. Side AC: from (0,0) to (6,8) This is the hypotenuse.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the triangle and its sides** Vertices: A = (0,0) B = (6,0) C = (6,8) Side AB: from (0,0) to (6,0) This is horizontal, length = 6 - 0 = 6. Side BC: from (6,0) to (6,8) This is vertical, length = 8 - 0 = 8. Side AC: from (0,0) to (6,8) This is the hypotenuse. Length = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10. --- **Step 2: Understand the squares** We construct a square on each side of the triangle, with that side as one side of the square. - Square on AB: side length 6 → area = 6^2 = 36 - Square on BC: side length 8 → area = 8^2 = 64 - Square on AC: side length 10 → area = 10^2 = 100 --- **Step 3: Total area of the three squares** Total area = 36 + 64 + 100 = 200 --- **Step 4: Final answer** The total area of the three squares is 200.

5. A construction crew is building a wheelchair ramp that must rise 1.5 meters vertically. The building code requires the ramp's horizontal run to be 12 meters. What is the exact length of the ramp's slanted surface in meters? Answer: 12.093 meters Solution: We are given a vertical rise of 1.5 meters and a horizontal run of 12 meters. We need the exact length of the slanted surface (the hypotenuse). Identify the right triangle.
   Full step-by-step solution

   We are given a vertical rise of 1.5 meters and a horizontal run of 12 meters. We need the exact length of the slanted surface (the hypotenuse). Step 1: Identify the right triangle. The vertical rise is one leg (a = 1.5 m), the horizontal run is the other leg (b = 12 m), and the ramp length is the hypotenuse (c). Step 2: Recall the Pythagorean theorem. It states: a^2 + b^2 = c^2. Step 3: Substitute the known values. a^2 = (1.5)^2 = 2.25 b^2 = (12)^2 = 144 So: 2.25 + 144 = c^2 Step 4: Add them. c^2 = 146.25 Step 5: Take the square root to find c. c = sqrt(146.25) Step 6: Simplify the square root. 146.25 = 14625/100 So c = sqrt(14625/100) = sqrt(14625) / 10 Step 7: Factor 14625 to simplify. 14625 ÷ 25 = 585, so 14625 = 25 × 585 585 ÷ 9 = 65, so 585 = 9 × 65 Thus 14625 = 25 × 9 × 65 = 225 × 65 So sqrt(14625) = sqrt(225 × 65) = sqrt(225) × sqrt(65) = 15 × sqrt(65) Step 8: Therefore: c = (15 × sqrt(65)) / 10 = (3 × sqrt(65)) / 2 Step 9: Numeric approximation. sqrt(65) ≈ 8.062257748 So c ≈ (3 × 8.062257748) / 2 = (24.186773244) / 2 = 12.093386622 Step 10: Round to three decimal places. c ≈ 12.093 meters Final answer: 12.093 meters

6. Liam is building a triangular garden bed with sides measuring 6 feet and 8 feet. He wants to place a diagonal support beam across the longest side. What length should the support beam be to ensure the garden bed forms a right triangle? Answer: 10 feet Solution: To find the length of the diagonal support beam, we need to determine the longest side of the right triangle. The problem states that the beam goes across the longest side, which is the hypotenuse. The two shorter sides of the triangular garden bed are 6 feet and 8 feet.
   Full step-by-step solution

   To find the length of the diagonal support beam, we need to determine the longest side of the right triangle. The problem states that the beam goes across the longest side, which is the hypotenuse. Step 1: Identify the given sides. The two shorter sides of the triangular garden bed are 6 feet and 8 feet. Step 2: Recall the Pythagorean theorem. For a right triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. The formula is: a^2 + b^2 = c^2 where a and b are the two shorter sides, and c is the hypotenuse. Step 3: Substitute the known values into the formula. Let a = 6 feet and b = 8 feet. So, 6^2 + 8^2 = c^2 Step 4: Calculate the squares. 6^2 = 36 8^2 = 64 Now add them: 36 + 64 = 100 So, c^2 = 100 Step 5: Find the length of the hypotenuse c. To find c, take the square root of c^2. c = square root of 100 c = 10 Step 6: State the final answer. The diagonal support beam should be 10 feet long.

7. (3² + 4²) = ? Answer: 25 Solution: Identify the operations inside the parentheses. We have (3² + 4²). This means we need to calculate 3 squared and 4 squared first.
   Full step-by-step solution

   Let's solve the problem step by step. Step 1: Identify the operations inside the parentheses. We have (3² + 4²). This means we need to calculate 3 squared and 4 squared first. Step 2: Calculate 3². 3² = 3 × 3 = 9. Step 3: Calculate 4². 4² = 4 × 4 = 16. Step 4: Add the results. 9 + 16 = 25. Step 5: Final answer. (3² + 4²) = 25.

8. A right triangle has legs measuring 12 cm and 16 cm. What is the length of the hypotenuse? Answer: 20 Solution: Identify the lengths of the legs: a = 12 cm, b = 16 cm. Substitute the values: 12² + 16² = c². Calculate the squares: 144 + 256 = c².
   Full step-by-step solution

   Step 1: Identify the lengths of the legs: a = 12 cm, b = 16 cm. Step 2: Apply the Pythagorean Theorem: a² + b² = c². Step 3: Substitute the values: 12² + 16² = c². Step 4: Calculate the squares: 144 + 256 = c². Step 5: Add the results: 400 = c². Step 6: Find the square root: c = sqrt(400) = 20. The length of the hypotenuse is 20 cm.

9. A drone is flying directly from its launch point to a delivery location. It travels 1.2 kilometers east and then 1.6 kilometers north. What is the straight-line distance, in kilometers, between the drone's launch point and its final delivery location? Answer: 2 Solution: The drone's path forms a right triangle. The eastward leg is 1.2 km, and the northward leg is 1.6 km. The straight-line distance is the hypotenuse.
   Full step-by-step solution

   Step 1: The drone's path forms a right triangle. The eastward leg is 1.2 km, and the northward leg is 1.6 km. The straight-line distance is the hypotenuse. Step 2: Apply the Pythagorean theorem: (leg1)^2 + (leg2)^2 = (hypotenuse)^2. Step 3: Substitute the known values: (1.2)^2 + (1.6)^2 = (distance)^2. Step 4: Calculate the squares: 1.44 + 2.56 = (distance)^2. Step 5: Add the results: 4.00 = (distance)^2. Step 6: Find the square root to solve for the distance: distance = sqrt(4.00). Step 7: sqrt(4.00) = 2. The straight-line distance is 2 kilometers.