Pythagorean Theorem

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Pythagorean Theorem · Grade 8 · Worksheet 2

1. A drone is flying from a control station to a delivery point. It travels 1.2 kilometers due east, then turns and flies 0.9 kilometers due north to reach its destination. What is the straight-line distance, in kilometers, from the control station to the delivery point? Round your answer to the nearest hundredth of a kilometer. Answer: 1.50 Solution: The drone's path forms a right triangle. The eastward leg is 1.2 km, and the northward leg is 0.9 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 0.9 km. The straight-line distance is the hypotenuse. Step 2: Apply the Pythagorean theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse. Step 3: Substitute the known values: (1.2)² + (0.9)² = c² Step 4: Calculate the squares: 1.44 + 0.81 = c² Step 5: Add the results: 2.25 = c² Step 6: Find the square root of both sides to solve for c: c = sqrt(2.25) Step 7: Calculate the square root: c = 1.5 Step 8: The problem asks for the answer rounded to the nearest hundredth, which is 1.50 km. The straight-line distance is 1.50 kilometers.

2. A right triangle has legs measuring 9 cm and 12 cm. What is the length of the hypotenuse? Answer: 15 Solution: Identify the lengths of the legs: a = 9 cm, b = 12 cm. Substitute the values: 9² + 12² = c². Calculate the squares: 81 + 144 = c².
   Full step-by-step solution

   Step 1: Identify the lengths of the legs: a = 9 cm, b = 12 cm. Step 2: Apply the Pythagorean Theorem: a² + b² = c². Step 3: Substitute the values: 9² + 12² = c². Step 4: Calculate the squares: 81 + 144 = c². Step 5: Add the results: 225 = c². Step 6: Find the square root: c = √225. Step 7: √225 = 15. The length of the hypotenuse is 15 cm.

3. Liam is designing a triangular garden with sides measuring 6 meters, 8 meters, and 10 meters. He wants to verify if this forms a right triangle by applying the Pythagorean theorem. Which side should be the hypotenuse, and does the garden form a right triangle? Answer: The hypotenuse should be 10 meters, and yes, it forms a right triangle because 6² + 8² = 10² (36 + 64 = 100). Solution: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) equals the sum of the squares of the lengths of the other two sides. That is: a² + b² = c², where c is the hypotenuse.
   Full step-by-step solution

   Step 1: Understand the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) equals the sum of the squares of the lengths of the other two sides. That is: a² + b² = c², where c is the hypotenuse. Step 2: Identify the longest side. The sides given are 6 m, 8 m, and 10 m. The longest side is 10 m, so if it is a right triangle, 10 m must be the hypotenuse. Step 3: Apply the theorem. Let a = 6 m, b = 8 m, and c = 10 m (hypotenuse candidate). Calculate: a² = 6² = 36 b² = 8² = 64 Sum: a² + b² = 36 + 64 = 100 c² = 10² = 100 Step 4: Compare results. Since a² + b² = 100 and c² = 100, the equation a² + b² = c² is true. Step 5: Conclusion. The hypotenuse should be 10 meters, and yes, it forms a right triangle because 6² + 8² = 10² (36 + 64 = 100).

4. Liam is building a triangular support brace for a bookshelf. The horizontal base measures 1.2 meters and the vertical side measures 0.9 meters. What is the length of the diagonal brace that connects the ends of these two sides? Round your answer to the nearest tenth of a meter. Answer: 1.5 Solution: We are finding the length of the diagonal brace in a right triangle. The horizontal base is 1.2 meters, and the vertical side is 0.9 meters. The diagonal is the hypotenuse.
   Full step-by-step solution

   We are finding the length of the diagonal brace in a right triangle. The horizontal base is 1.2 meters, and the vertical side is 0.9 meters. The diagonal is the hypotenuse. Step 1: Recall the Pythagorean theorem: a^2 + b^2 = c^2 where a and b are the two perpendicular sides, and c is the hypotenuse. Step 2: Assign values: a = 1.2, b = 0.9, c = ? Step 3: Apply the formula: c^2 = (1.2)^2 + (0.9)^2 Step 4: Calculate squares: 1.2^2 = 1.44 0.9^2 = 0.81 Step 5: Add them: 1.44 + 0.81 = 2.25 Step 6: Take the square root: c = sqrt(2.25) c = 1.5 Step 7: Round to the nearest tenth: 1.5 is already to the nearest tenth. Final answer: 1.5 meters.

5. A right triangle has legs of length 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 of both sides to solve for c: c = √400. Step 7: √400 = 20. The length of the hypotenuse is 20 cm.

6. √(20² + 15²) = ? Answer: 25 Solution: Calculate 20 squared: 20² = 400 Calculate 15 squared: 15² = 225 Add the two squared values: 400 + 225 = 625 Take the square root of the sum: √625 = 25 The answer is 25.
   Full step-by-step solution

   Step 1: Calculate 20 squared: 20² = 400 Step 2: Calculate 15 squared: 15² = 225 Step 3: Add the two squared values: 400 + 225 = 625 Step 4: Take the square root of the sum: √625 = 25 The answer is 25.

7. Liam is designing a triangular garden with sides measuring 6 meters and 8 meters. He needs to know the length of the diagonal side to purchase enough fencing. What is the length of the diagonal side of his garden? Answer: 10 Solution: 1. The "diagonal side" means the third side opposite the right angle if this is a right triangle. Often, 6 and 8 are the two perpendicular sides, and the diagonal is the hypotenuse.
   Full step-by-step solution

   Let's go step by step. 1. Understand the problem: Liam has a triangular garden with two sides given: 6 meters and 8 meters. The "diagonal side" means the third side opposite the right angle if this is a right triangle. Often, 6 and 8 are the two perpendicular sides, and the diagonal is the hypotenuse. 2. Check if it's a right triangle: The problem doesn't explicitly say it's a right triangle, but the numbers 6 and 8 are common in a 3-4-5 right triangle scaled by 2: 3-4-5 triangle → multiply by 2 → 6-8-10. So the diagonal would be 10 if it's a right triangle. 3. Apply the Pythagorean theorem: For a right triangle: a^2 + b^2 = c^2 Let a = 6, b = 8, c = diagonal. Then: 6^2 + 8^2 = c^2 36 + 64 = c^2 100 = c^2 4. Solve for c: c = square root of 100 c = 10 5. Conclusion: The diagonal side is 10 meters. Answer: 10