Pythagorean 2D

lessonbunny.com

Answer Key & Explanations

Pythagorean 2D · Grade 8 · Worksheet 1

1. √(37² - 12²) = ? Answer: 35 Solution: Write the Pythagorean theorem: a² + b² = c² We have c = 37 and b = 12, so we need to find a: a² + 12² = 37² Calculate the squares: a² + 144 = 1369 Subtract 144 from both sides: a² = 1369 - 144 Simplify: a² = 1225 Take the square root: a = √1225 √1225 = 35 The answer is 35.
   Full step-by-step solution

   Step 1: Write the Pythagorean theorem: a² + b² = c² Step 2: We have c = 37 and b = 12, so we need to find a: a² + 12² = 37² Step 3: Calculate the squares: a² + 144 = 1369 Step 4: Subtract 144 from both sides: a² = 1369 - 144 Step 5: Simplify: a² = 1225 Step 6: Take the square root: a = √1225 Step 7: √1225 = 35 The answer is 35.

2. Liam is building a triangular support brace for his bookshelf. The brace will be a right triangle where the horizontal side measures 48 cm and the vertical side measures 64 cm. What is the length of the diagonal brace that Liam needs to cut? Answer: 80 cm Solution: We are given a right triangle with the horizontal side = 48 cm and the vertical side = 64 cm. We need to find the length of the diagonal brace, which is the hypotenuse. Recall the Pythagorean theorem.
   Full step-by-step solution

   We are given a right triangle with the horizontal side = 48 cm and the vertical side = 64 cm. We need to find the length of the diagonal brace, which is the hypotenuse. Step 1: Recall the Pythagorean theorem. For a right triangle with legs a and b and hypotenuse c: a^2 + b^2 = c^2. Step 2: Assign values. Let a = 48 cm, b = 64 cm, and c = diagonal length. So: 48^2 + 64^2 = c^2. Step 3: Calculate squares. 48^2 = 2304 64^2 = 4096 Step 4: Add the squares. 2304 + 4096 = 6400 Step 5: Now we have: c^2 = 6400 Step 6: Find c by taking the square root. c = sqrt(6400) c = 80 Step 7: Conclusion. The diagonal brace Liam needs to cut is 80 cm long. Final answer: 80 cm

3. Liam is building a rectangular garden in his backyard. The garden measures 12 feet by 5 feet. He wants to install a diagonal stone path from one corner of the garden to the opposite corner. How long will the stone path be? Round your answer to the nearest tenth of a foot. Answer: 13.0 Solution: We are given a rectangular garden that is 12 feet by 5 feet. We need the length of the diagonal from one corner to the opposite corner.
   Full step-by-step solution

   We are given a rectangular garden that is 12 feet by 5 feet. We need the length of the diagonal from one corner to the opposite corner. Step 1: Recognize that the diagonal of a rectangle forms a right triangle with the length and width as the two legs. The diagonal is the hypotenuse. Step 2: Apply the Pythagorean theorem: a^2 + b^2 = c^2 where a = 12 ft, b = 5 ft, and c = diagonal length. Step 3: Substitute the values: 12^2 + 5^2 = c^2 144 + 25 = c^2 169 = c^2 Step 4: Solve for c: c = square root of 169 c = 13 Step 5: The problem says to round to the nearest tenth of a foot. 13 is already 13.0 when rounded to the nearest tenth. Final answer: The stone path will be 13.0 feet long.

4. Olivia is installing a diagonal support beam in a rectangular gate she is building. The gate is 15 feet wide and 20 feet tall. She needs to cut a beam that goes from the bottom left corner to the top right corner of the gate. How long must the diagonal support beam be in feet? Answer: 25 Solution: The width (15 feet) and height (20 feet) are the legs of a right triangle, and the diagonal beam is the hypotenuse.
   Full step-by-step solution

   Step 1: The width (15 feet) and height (20 feet) are the legs of a right triangle, and the diagonal beam is the hypotenuse. Step 2: Apply the Pythagorean theorem: a² + b² = c², where a = 15 and b = 20. Step 3: Calculate 15² = 225 Step 4: Calculate 20² = 400 Step 5: Add the squares: 225 + 400 = 625 Step 6: Find the square root: sqrt(625) = 25 The diagonal support beam must be 25 feet long.

5. √(15² + 20²) = ? Answer: 25 Solution: √(15² + 20²) Calculate 15 squared 15 × 15 = 225 Calculate 20 squared 20 × 20 = 400 225 + 400 = 625 √625 = ? We know 25 × 25 = 625, so √625 = 25. The result is 25.
   Full step-by-step solution

   Let's solve step by step. We are given: √(15² + 20²) **Step 1: Calculate 15 squared** 15 × 15 = 225 **Step 2: Calculate 20 squared** 20 × 20 = 400 **Step 3: Add the two results** 225 + 400 = 625 **Step 4: Take the square root** √625 = ? We know 25 × 25 = 625, so √625 = 25. **Step 5: Final answer** The result is 25. **Reasoning:** This is an application of the Pythagorean theorem formula for the hypotenuse: c = √(a² + b²) Here a = 15, b = 20, so c = √(225 + 400) = √625 = 25.

6. A research drone is flying at an altitude of 1.2 × 10³ meters. It detects a research station on the ground that is 1.6 × 10³ meters away horizontally from the drone's position directly above its launch point. What is the straight-line distance in meters between the drone and the research station? Express your answer in standard form. Answer: 2000 Solution: Identify the right triangle sides. The altitude is the vertical leg: 1.2 × 10³ = 1200 meters. The horizontal distance is the other leg: 1.6 × 10³ = 1600 meters.
   Full step-by-step solution

   Step 1: Identify the right triangle sides. The altitude is the vertical leg: 1.2 × 10³ = 1200 meters. The horizontal distance is the other leg: 1.6 × 10³ = 1600 meters. The straight-line distance is the hypotenuse. Step 2: Apply the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Step 3: Substitute the values: 1200² + 1600² = c² Step 4: Calculate: 1200² = 1,440,000 and 1600² = 2,560,000 Step 5: Add: 1,440,000 + 2,560,000 = 4,000,000 Step 6: Take the square root: c = sqrt(4,000,000) = 2000 Step 7: The straight-line distance is 2000 meters, which in standard form is 2000. The answer is 2000.

7. √(12² + 16²) = ? Answer: 20 Solution: We are given: √(12² + 16²) Calculate the squares inside the square root. 12² = 12 × 12 = 144 16² = 16 × 16 = 256 Add the two results. 144 + 256 = 400 Take the square root of the sum.
   Full step-by-step solution

   We are given: √(12² + 16²) Step 1: Calculate the squares inside the square root. 12² = 12 × 12 = 144 16² = 16 × 16 = 256 Step 2: Add the two results. 144 + 256 = 400 Step 3: Take the square root of the sum. √400 = 20 So the final answer is 20. Explanation: This expression is the Pythagorean formula for the hypotenuse of a right triangle with legs 12 and 16. The calculation shows that the hypotenuse length is 20.