Pythagorean Theorem

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Pythagorean Theorem · Grade 8 · Worksheet 1

1. A rectangular field has a length of 3.6 × 10³ meters and a width of 4.8 × 10³ meters. A diagonal path is built across the field. Using the Pythagorean theorem, find the length of the diagonal path in scientific notation. Answer: 6.0×10³ Solution: The diagonal of a rectangle forms a right triangle with the length and width as the legs.
   Full step-by-step solution

   Step 1: The diagonal of a rectangle forms a right triangle with the length and width as the legs. Step 2: Apply the Pythagorean theorem: diagonal² = length² + width² Step 3: Substitute the values: diagonal² = (3.6 × 10³)² + (4.8 × 10³)² Step 4: Calculate (3.6 × 10³)² = 3.6² × (10³)² = 12.96 × 10⁶ Step 5: Calculate (4.8 × 10³)² = 4.8² × (10³)² = 23.04 × 10⁶ Step 6: Add the results: 12.96 × 10⁶ + 23.04 × 10⁶ = 36.00 × 10⁶ Step 7: Take the square root: diagonal = sqrt(36.00 × 10⁶) = sqrt(36.00) × sqrt(10⁶) = 6.00 × 10³ Step 8: Write in proper scientific notation: 6.0 × 10³ The answer is 6.0×10³.

2. 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.

3. Liam is building a triangular support brace for his bookshelf. The brace will be a right triangle where the two shorter sides measure 1.2 meters and 1.6 meters. What is the length of the longest side (the hypotenuse) of this triangular brace? Answer: 2 Solution: We are given a right triangle with the two shorter sides measuring 1.2 meters and 1.6 meters. We need to find the length of the longest side (the hypotenuse).
   Full step-by-step solution

   We are given a right triangle with the two shorter sides measuring 1.2 meters and 1.6 meters. We need to find the length of the longest side (the hypotenuse). Step 1: Recall the Pythagorean theorem For a right triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse. If the sides are a, b, and the hypotenuse is c, then: a^2 + b^2 = c^2 Step 2: Substitute the given values Here, a = 1.2, b = 1.6 So: (1.2)^2 + (1.6)^2 = c^2 Step 3: Calculate the squares 1.2^2 = 1.44 1.6^2 = 2.56 Step 4: Add them 1.44 + 2.56 = 4.00 Step 5: Now we have c^2 = 4 Step 6: Take the square root of both sides c = sqrt(4) = 2 Step 7: Conclusion The length of the longest side (hypotenuse) is 2 meters. Final answer: 2

4. A right triangle has legs measuring 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.

5. Charlotte is designing a square garden with a side length of 7 meters. She wants to create a diagonal pathway that goes from one corner of the garden to the opposite corner. To prove that the length of the diagonal is exactly 7√2 meters, Charlotte decides to use an area model. She draws a large square with side length (a + b) where a = 7 meters and b = 7 meters. Inside, she places four congruent right triangles (each with legs a and b) and a smaller square with side length c (the diagonal). Using the area model, explain why the Pythagorean theorem (a² + b² = c²) holds true for this garden, and find the value of c². Answer: 98 square meters Solution: The large square has side length (a + b) = 7 + 7 = 14 meters. Its area is (14)² = 196 square meters. Each of the four right triangles has legs a = 7 meters and b = 7 meters.
   Full step-by-step solution

   Step 1: The large square has side length (a + b) = 7 + 7 = 14 meters. Its area is (14)² = 196 square meters. Step 2: Each of the four right triangles has legs a = 7 meters and b = 7 meters. The area of one triangle is (1/2) * a * b = (1/2) * 7 * 7 = 24.5 square meters. The total area of all four triangles is 4 * 24.5 = 98 square meters. Step 3: The smaller square in the center has side length c (the diagonal of the original garden). Its area is c². Step 4: The area of the large square equals the area of the four triangles plus the area of the small square: (a + b)² = 4 * (1/2 * a * b) + c². Step 5: Substitute a = 7 and b = 7: 196 = 98 + c². Step 6: Solve for c²: c² = 196 - 98 = 98 square meters. Step 7: This shows that a² + b² = 49 + 49 = 98 = c², proving the Pythagorean theorem. The value of c² is 98 square meters.

6. 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.