Pythagorean Theorem

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Pythagorean Theorem · Grade 8 · Worksheet 3

1. 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.

2. √(8² + 15²) = ? Answer: 17 Solution: Calculate 8 squared: 8² = 64 Calculate 15 squared: 15² = 225 Add the two squared values: 64 + 225 = 289 Take the square root of the sum: √289 = 17 The answer is 17.
   Full step-by-step solution

   Step 1: Calculate 8 squared: 8² = 64 Step 2: Calculate 15 squared: 15² = 225 Step 3: Add the two squared values: 64 + 225 = 289 Step 4: Take the square root of the sum: √289 = 17 The answer is 17.

3. Emma is flying a kite that is 120 feet above the ground. Her friend Noah is standing 50 feet away from the point directly below the kite. If the kite string is pulled taut in a straight line, what is the length of the kite string from Emma's hand to the kite? Answer: 130 Solution: Identify the right triangle. The vertical side is the kite's height (120 ft). The horizontal side is Noah's distance from the point below the kite (50 ft).
   Full step-by-step solution

   Step 1: Identify the right triangle. The vertical side is the kite's height (120 ft). The horizontal side is Noah's distance from the point below the kite (50 ft). The hypotenuse is the kite string length we need to find. Step 2: Apply the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Step 3: Substitute the known values: 120² + 50² = c² Step 4: Calculate the squares: 14400 + 2500 = c² Step 5: Add the results: 16900 = c² Step 6: Take the square root of both sides: c = sqrt(16900) = 130 Step 7: The length of the kite string is 130 feet.

4. A drone is flying from a park bench to the top of a flagpole. The drone takes off from a bench that is 24 meters from the base of the flagpole. The flagpole itself is 7 meters tall. What is the straight-line distance, in meters, that the drone flies from the bench to the top of the flagpole? Answer: 25 Solution: Identify the sides of the right triangle. The ground distance from the bench to the flagpole is one leg (24 m). The height of the flagpole is the other leg (7 m).
   Full step-by-step solution

   Step 1: Identify the sides of the right triangle. The ground distance from the bench to the flagpole is one leg (24 m). The height of the flagpole is the other leg (7 m). The drone's flight path is the hypotenuse. Step 2: Apply the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Step 3: Substitute the known values: 24² + 7² = c² Step 4: Calculate the squares: 576 + 49 = c² Step 5: Add the results: 625 = c² Step 6: Find the square root to solve for c: c = sqrt(625) = 25 Step 7: The straight-line distance the drone flies is 25 meters.

5. A drone is flying from a control station to a delivery point. It travels 2.4 kilometers due east, then makes a sharp turn and travels 3.2 kilometers due north to reach its destination. What is the straight-line distance, in kilometers, from the control station to the delivery point? Answer: 4 Solution: The drone's path forms a right triangle. The eastward leg is 2.4 km, and the northward leg is 3.2 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 2.4 km, and the northward leg is 3.2 km. The straight-line distance is the hypotenuse. Step 2: Apply the Pythagorean theorem: a^2 + b^2 = c^2, where a = 2.4 and b = 3.2. Step 3: Calculate a^2: 2.4^2 = 5.76 Step 4: Calculate b^2: 3.2^2 = 10.24 Step 5: Add the squares: 5.76 + 10.24 = 16 Step 6: Find the square root to get c: c = sqrt(16) = 4 Step 7: The straight-line distance is 4 kilometers.

6. 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 the side of the triangle serving as one side of the square. What is the total area of the two smaller squares combined? Answer: 100 Solution: A = (0, 0) B = (6, 0) C = (6, 8) This is a right triangle with the right angle at B = (6, 0) because AB is horizontal and BC is vertical.
   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) This is a right triangle with the right angle at B = (6, 0) because AB is horizontal and BC is vertical. --- **Step 2: Find the side lengths** Side AB: from (0,0) to (6,0) Length = 6 Side BC: from (6,0) to (6,8) Length = 8 Side AC: from (0,0) to (6,8) Length = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10 --- **Step 3: Interpret the problem** A square is constructed on each side of the triangle, with the triangle's side as one side of the square. So: - Square on AB has area = 6^2 = 36 - Square on BC has area = 8^2 = 64 - Square on AC has area = 10^2 = 100 --- **Step 4: Identify the "two smaller squares"** The sides of the triangle are length 6, 8, and 10. The two smaller sides are 6 and 8. The two smaller squares are the ones on sides AB and BC. --- **Step 5: Add their areas** Area of square on AB = 36 Area of square on BC = 64 Total = 36 + 64 = 100 --- **Final Answer:** 100

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 right triangle has legs of length 5 cm and 12 cm. What is the length of the hypotenuse? Answer: 13 Solution: Write the Pythagorean theorem: a² + b² = c² Substitute the given values: 5² + 12² = c² Calculate the squares: 25 + 144 = c² Add the results: 169 = c² Take the square root of both sides: c = √169 Calculate the square root: c = 13 The length of the hypotenuse is 13 cm.
   Full step-by-step solution

   Step 1: Write the Pythagorean theorem: a² + b² = c² Step 2: Substitute the given values: 5² + 12² = c² Step 3: Calculate the squares: 25 + 144 = c² Step 4: Add the results: 169 = c² Step 5: Take the square root of both sides: c = √169 Step 6: Calculate the square root: c = 13 The length of the hypotenuse is 13 cm.