Pythagorean Theorem

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Pythagorean Theorem · Grade 8 · Worksheet 3

1. A right triangle has legs measuring 7 cm and 24 cm. Using the Pythagorean theorem, find the length of the hypotenuse in centimeters. Answer: 25 Solution: Recall the Pythagorean theorem. For a right triangle with legs a and b and hypotenuse c, the formula is: a squared + b squared = c squared. Identify the given lengths.
   Full step-by-step solution

   Step 1: Recall the Pythagorean theorem. For a right triangle with legs a and b and hypotenuse c, the formula is: a squared + b squared = c squared. Step 2: Identify the given lengths. Here, leg a = 7 cm and leg b = 24 cm. We are solving for the hypotenuse c. Step 3: Substitute the known values into the formula. 7 squared + 24 squared = c squared. Step 4: Calculate the squares. 7 squared = 7 * 7 = 49 24 squared = 24 * 24 = 576 Step 5: Add the squares. 49 + 576 = 625 Step 6: Now we have: c squared = 625 Step 7: To find c, take the square root of both sides. c = square root of 625 Step 8: Since 25 * 25 = 625, the square root of 625 is 25. Step 9: Conclusion. The length of the hypotenuse is 25 cm.

2. A right triangle has legs of length 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 to solve for c: c = sqrt(225) = 15. The length of the hypotenuse is 15 cm.

3. 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.

4. √(28² + 45²) = ? Answer: 53 Solution: Calculate 28 squared: 28² = 784 Calculate 45 squared: 45² = 2025 Add the two squared values: 784 + 2025 = 2809 Take the square root of the sum: √2809 = 53 The answer is 53.
   Full step-by-step solution

   Step 1: Calculate 28 squared: 28² = 784 Step 2: Calculate 45 squared: 45² = 2025 Step 3: Add the two squared values: 784 + 2025 = 2809 Step 4: Take the square root of the sum: √2809 = 53 The answer is 53.

5. Lena is designing a triangular garden plot with sides measuring 6 meters and 8 meters. She wants to install a decorative fence along the longest side. To determine how much fencing material to buy, she needs to calculate the length of this third side. What is the length of the longest side of Lena's garden plot? Answer: 10 meters Solution: We are told Lena has a triangular garden plot with sides 6 m and 8 m, and she needs the length of the longest side.
   Full step-by-step solution

   We are told Lena has a triangular garden plot with sides 6 m and 8 m, and she needs the length of the longest side. Since the problem doesn’t specify the angle between these sides, the most common assumption for such a problem is that the triangle is a right triangle. This is because the numbers 6 and 8 often appear in a 3-4-5 right triangle multiple. Step 1: Identify the sides Let the two given sides be 6 m and 8 m. If the triangle is right-angled, the longest side will be the hypotenuse. Step 2: Apply the Pythagorean theorem The Pythagorean theorem states: a^2 + b^2 = c^2 where c is the hypotenuse. Let a = 6, b = 8, then: 6^2 + 8^2 = c^2 36 + 64 = c^2 100 = c^2 Step 3: Solve for c c^2 = 100 c = square root of 100 c = 10 Step 4: Conclusion The longest side is 10 meters. This matches the typical 6-8-10 right triangle, which is consistent with the given numbers. Answer: 10 meters

6. 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.

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,4). A square is constructed on the hypotenuse of this triangle. What is the area of this square? Answer: 52 Solution: A = (0,0) B = (6,0) C = (6,4) Side AB: from (0,0) to (6,0) → length = 6 (horizontal) Side BC: from (6,0) to (6,4) → length = 4 (vertical) Side AC: from (0,0) to (6,4) → hypotenuse AC^2 = AB^2 + BC^2 AC^2 = 6^2 + 4^2 AC^2 = 36 + 16 AC^2 = 52 AC = sqrt(52) The square is constructed on the…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the triangle's sides** Vertices: A = (0,0) B = (6,0) C = (6,4) Side AB: from (0,0) to (6,0) → length = 6 (horizontal) Side BC: from (6,0) to (6,4) → length = 4 (vertical) Side AC: from (0,0) to (6,4) → hypotenuse --- **Step 2: Find the hypotenuse length** Using the Pythagorean theorem: AC^2 = AB^2 + BC^2 AC^2 = 6^2 + 4^2 AC^2 = 36 + 16 AC^2 = 52 AC = sqrt(52) --- **Step 3: Understand the problem** The square is constructed on the hypotenuse AC. That means AC is one side of the square. --- **Step 4: Find the area of the square** Area of square = (side length)^2 Side length = AC = sqrt(52) Area = (sqrt(52))^2 = 52 --- **Step 5: Conclusion** The area of the square is 52. --- **Final answer:** 52

8. A drone is flying at a constant altitude of 120 meters. Its controller on the ground is 50 meters away from the point directly below the drone. What is the straight-line distance between the drone and its controller? Answer: 130 Solution: The drone's altitude of 120 meters forms one leg of a right triangle. The horizontal distance of 50 meters forms the other leg of the right triangle.
   Full step-by-step solution

   Step 1: The drone's altitude of 120 meters forms one leg of a right triangle. Step 2: The horizontal distance of 50 meters forms the other leg of the right triangle. Step 3: Apply the Pythagorean theorem: a² + b² = c² Step 4: Substitute the values: 120² + 50² = c² Step 5: Calculate: 14400 + 2500 = c² Step 6: Add: 16900 = c² Step 7: Take the square root: c = sqrt(16900) Step 8: Simplify: c = 130 The straight-line distance is 130 meters.

9. (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.