Pythagorean Theorem

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Pythagorean Theorem · Grade 8 · Worksheet 2

1. Noah is helping his teacher create a visual proof of the Pythagorean theorem for a classroom display. He draws a large square with side length (a + b), where a = 6 cm and b = 1 cm. Inside the large square, Noah places four congruent right triangles, each with legs a and b, arranged so that their hypotenuses form a smaller inner square. Explain how this area model proves that a² + b² = c² for any right triangle, and determine the side length c of the inner square (the hypotenuse). Answer: sqrt(37) or approximately 6.08 cm Solution: The large square has side length (a + b) = 6 + 1 = 7 cm. Its area is (a + b)² = 7² = 49 cm². Each right triangle has legs a = 6 cm and b = 1 cm.
   Full step-by-step solution

   Step 1: The large square has side length (a + b) = 6 + 1 = 7 cm. Its area is (a + b)² = 7² = 49 cm². Step 2: Each right triangle has legs a = 6 cm and b = 1 cm. The area of one triangle is (1/2) × a × b = (1/2) × 6 × 1 = 3 cm². Four triangles have total area 4 × 3 = 12 cm². Step 3: The inner square has side length c (the hypotenuse of each right triangle). Its area is c². Step 4: The area of the large square can also be expressed as the sum of the areas of the four triangles and the inner square: c² + 12 cm². Step 5: Equate the two expressions for the large square's area: (a + b)² = c² + 4 × (1/2)ab, or 49 = c² + 12. Step 6: Expand (a + b)² = a² + 2ab + b². So a² + 2ab + b² = c² + 2ab. Subtract 2ab from both sides to get a² + b² = c². Step 7: Using a = 6 and b = 1, we have 6² + 1² = 36 + 1 = 37 = c². Therefore, c = sqrt(37) cm. Step 8: Numerically, sqrt(37) is approximately 6.08 cm. The inner square area is 37 cm², and 49 = 37 + 12 confirms the arrangement. Thus, the area model proves that a² + b² = c² for any right triangle, and for this specific triangle, the hypotenuse c = sqrt(37) cm.

2. A drone is flying directly from a control station to a delivery point. The drone flies 2.4 kilometers east and then 1.8 kilometers north to reach its destination. What is the straight-line distance, in kilometers, between the control station and the delivery point? Answer: 3 Solution: The drone's path forms a right triangle where the eastward distance (2.4 km) and northward distance (1.8 km) are the two shorter sides.
   Full step-by-step solution

   Step 1: The drone's path forms a right triangle where the eastward distance (2.4 km) and northward distance (1.8 km) are the two shorter sides. Step 2: Apply the Pythagorean theorem: a² + b² = c², where c is the straight-line distance. Step 3: Substitute the known values: (2.4)² + (1.8)² = c² Step 4: Calculate the squares: 5.76 + 3.24 = c² Step 5: Add the results: 9.00 = c² Step 6: Take the square root of both sides: c = sqrt(9.00) Step 7: c = 3 The straight-line distance is 3 kilometers.

3. 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

4. A right triangle has legs measuring 6 cm and 8 cm. What is the length of the hypotenuse? Answer: 10 cm Solution: We are given a right triangle with legs 6 cm and 8 cm. We need to find the length of the hypotenuse. Recall the Pythagorean theorem.
   Full step-by-step solution

   We are given a right triangle with legs 6 cm and 8 cm. We need to find the length of the hypotenuse. Step 1: Recall the Pythagorean theorem. For a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. If the legs are a and b, and the hypotenuse is c, then: a^2 + b^2 = c^2 Step 2: Substitute the given values into the formula. Here, a = 6 cm and b = 8 cm. So: 6^2 + 8^2 = c^2 Step 3: Calculate the squares. 6^2 = 36 8^2 = 64 So: 36 + 64 = c^2 Step 4: Add the results. 36 + 64 = 100 Therefore: c^2 = 100 Step 5: Find c by taking the square root. c = square root of 100 c = 10 Step 6: State the final answer. The length of the hypotenuse is 10 cm.

5. Isabella is creating a geometric art piece using congruent right triangles. She cuts out four identical right triangles, each with legs measuring 3 inches and 4 inches. She arranges them around a central square to form a larger square, as shown in a classic proof of the Pythagorean theorem. Explain how this arrangement proves that for any right triangle with legs a and b and hypotenuse c, the relationship a² + b² = c² holds true. Answer: The area of the large square can be expressed both as (a+b)² and as c² + 2ab. Setting these equal and simplifying gives a² + b² = c², proving the theorem. Solution: Let the legs of each right triangle be a = 3 and b = 4. The hypotenuse is c. The triangles are arranged so that their hypotenuses form the sides of the inner square.
   Full step-by-step solution

   Step 1: Let the legs of each right triangle be a = 3 and b = 4. The hypotenuse is c. The triangles are arranged so that their hypotenuses form the sides of the inner square. The side length of the large square is a + b = 3 + 4 = 7 inches. Step 2: Calculate the area of the large square using its side length: Area = (a+b)² = (7)² = 49 square inches. Step 3: Calculate the area of the large square by adding the areas of its parts. Each triangle has area (1/2)ab = (1/2)(3)(4) = 6 square inches. Four triangles have total area 4 × 6 = 24 square inches. The inner square has side length c, so its area is c². Step 4: The total area is also c² + 24 square inches. Equate the two expressions: (a+b)² = c² + 4(1/2)ab, or 49 = c² + 24. Step 5: Expand (a+b)² = a² + 2ab + b². So a² + 2ab + b² = c² + 2ab. Subtract 2ab from both sides: a² + b² = c². Step 6: Check with numbers: 3² + 4² = 9 + 16 = 25, and c = sqrt(25) = 5. The inner square area is 25 square inches, so 49 = 25 + 24 holds true. Thus, the arrangement proves that a² + b² = c² for any right triangle.

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 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.

7. √(12² + 35²) = ? Answer: 37 Solution: Calculate 12 squared: 12² = 144 Calculate 35 squared: 35² = 1225 Add the two squared values: 144 + 1225 = 1369 Take the square root of the sum: √1369 = 37 The answer is 37.
   Full step-by-step solution

   Step 1: Calculate 12 squared: 12² = 144 Step 2: Calculate 35 squared: 35² = 1225 Step 3: Add the two squared values: 144 + 1225 = 1369 Step 4: Take the square root of the sum: √1369 = 37 The answer is 37.