Pythagorean 2D

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Pythagorean 2D · Grade 8 · Worksheet 3

1. Liam is building a triangular support brace for his bookshelf. The brace will be a right triangle where the base measures 2.4 meters and the height measures 0.7 meters. What is the length of the diagonal support beam, in meters? Round your answer to the nearest hundredth. Answer: 2.50 Solution: We are given a right triangle with base = 2.4 m and height = 0.7 m. We need to find the length of the diagonal support beam, which is the hypotenuse. a^2 + b^2 = c^2 where a and b are the legs, and c is the hypotenuse.
   Full step-by-step solution

   We are given a right triangle with base = 2.4 m and height = 0.7 m. We need to find the length of the diagonal support beam, which is the hypotenuse. Step 1: Recall the Pythagorean theorem For a right triangle: a^2 + b^2 = c^2 where a and b are the legs, and c is the hypotenuse. Here, base = 2.4 m, height = 0.7 m, so: a = 2.4, b = 0.7, c = ? Step 2: Substitute into the formula c^2 = (2.4)^2 + (0.7)^2 Step 3: Calculate squares (2.4)^2 = 5.76 (0.7)^2 = 0.49 Step 4: Add them c^2 = 5.76 + 0.49 c^2 = 6.25 Step 5: Take the square root c = sqrt(6.25) c = 2.5 Step 6: Round to the nearest hundredth 2.5 is already 2.50 to the nearest hundredth. Final answer: The diagonal support beam is 2.50 meters long.

2. Sophia is building a rectangular skateboard ramp in her driveway. The ramp has a horizontal base that is 36 inches long and a vertical height of 16 inches. She needs to cut a diagonal support piece that connects the top of the vertical height to the far end of the horizontal base. How long, in inches, should Sophia cut the diagonal support piece? Answer: 39.4 Solution: The horizontal base (36 inches) and vertical height (16 inches) are the two legs of a right triangle. The diagonal support piece is the hypotenuse of this right triangle.
   Full step-by-step solution

   Step 1: The horizontal base (36 inches) and vertical height (16 inches) are the two legs of a right triangle. Step 2: The diagonal support piece is the hypotenuse of this right triangle. Step 3: Apply the Pythagorean theorem: a^2 + b^2 = c^2, where a = 36 and b = 16. Step 4: Calculate 36^2 = 1296 Step 5: Calculate 16^2 = 256 Step 6: Add the squares: 1296 + 256 = 1552 Step 7: Find the square root: sqrt(1552) = 39.395... Step 8: Round to the nearest tenth: 39.4 The diagonal support piece must be 39.4 inches long.

3. Emma is flying a kite from a string that is 85 meters long. The kite is directly above a point on the ground that is 36 meters away from where Emma is standing. How high is the kite flying above the ground, in meters? Answer: 77 Solution: Identify the right triangle. The hypotenuse is the kite string (85 m). One leg is the horizontal distance from Emma to the point below the kite (36 m).
   Full step-by-step solution

   Step 1: Identify the right triangle. The hypotenuse is the kite string (85 m). One leg is the horizontal distance from Emma to the point below the kite (36 m). The other leg is the height we need to find. Step 2: Apply the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Step 3: Let h be the height. So, h² + 36² = 85². Step 4: Calculate the squares: h² + 1296 = 7225. Step 5: Isolate h²: h² = 7225 - 1296. Step 6: Subtract: h² = 5929. Step 7: Find the square root: h = sqrt(5929). Step 8: Calculate: h = 77. The kite is flying 77 meters above the ground.

4. √(20² - 12²) = ? Answer: 16 Solution: Write the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Let the unknown leg be x.
   Full step-by-step solution

   Step 1: Write the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Step 2: In this problem, we have c = 20 and one leg = 12. Let the unknown leg be x. Step 3: Set up the equation: x² + 12² = 20² Step 4: Calculate the squares: x² + 144 = 400 Step 5: Subtract 144 from both sides: x² = 400 - 144 Step 6: Simplify: x² = 256 Step 7: Take the square root of both sides: x = √256 Step 8: Calculate: x = 16 The answer is 16.

5. Olivia is helping her family set up a rectangular trampoline in their backyard. The trampoline measures 9 feet wide and 15 feet long. To ensure the frame is perfectly square, Olivia's dad suggests measuring the diagonal distance from one corner to the opposite corner. What is the length of this diagonal, in feet? Round your answer to the nearest tenth of a foot. Answer: 17.5 Solution: The width (9 ft) and length (15 ft) are the legs of a right triangle, and the diagonal is the hypotenuse. Calculate 9^2 = 81. Calculate 15^2 = 225.
   Full step-by-step solution

   Step 1: The width (9 ft) and length (15 ft) are the legs of a right triangle, and the diagonal is the hypotenuse. Step 2: Apply the Pythagorean theorem: a^2 + b^2 = c^2, where a = 9 and b = 15. Step 3: Calculate 9^2 = 81. Step 4: Calculate 15^2 = 225. Step 5: Add the squares: 81 + 225 = 306. Step 6: Find the square root: sqrt(306) = 17.4928... Step 7: Round to the nearest tenth: 17.5. The diagonal of the trampoline is 17.5 feet.

6. A right triangle has legs of length 7 cm and 24 cm. What is the length of the hypotenuse? Answer: 25 cm Solution: We are given a right triangle with legs of length 7 cm and 24 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 of length 7 cm and 24 cm. We need to find the length of the hypotenuse. Step 1: Recall the Pythagorean theorem. For a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. If the legs are a and b, and the hypotenuse is c, then: a^2 + b^2 = c^2 Step 2: Substitute the given lengths. Here, a = 7 cm, b = 24 cm. So: 7^2 + 24^2 = c^2 Step 3: Calculate the squares. 7^2 = 49 24^2 = 576 So: 49 + 576 = c^2 Step 4: Add the results. 49 + 576 = 625 Therefore: c^2 = 625 Step 5: Find c by taking the square root. c = square root of 625 Since 25 * 25 = 625, we have: c = 25 Step 6: State the final answer. The length of the hypotenuse is 25 cm.

7. Liam is building a triangular support brace for a bookshelf. The vertical side of the triangle is 1.2 meters long, and the horizontal side is 0.5 meters long. What is the length of the diagonal brace that Liam needs to cut? Round your answer to the nearest tenth of a meter. Answer: 1.3 Solution: We are given a right triangle with vertical side 1.2 meters and horizontal side 0.5 meters. The diagonal brace is the hypotenuse.
   Full step-by-step solution

   We are given a right triangle with vertical side 1.2 meters and horizontal side 0.5 meters. The diagonal brace 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: a = 1.2 m (vertical side) b = 0.5 m (horizontal side) c = ? (diagonal brace) Step 3: Substitute into the formula: (1.2)^2 + (0.5)^2 = c^2 Step 4: Calculate squares: 1.2^2 = 1.44 0.5^2 = 0.25 Step 5: Add them: 1.44 + 0.25 = 1.69 Step 6: So c^2 = 1.69 Step 7: Take the square root: c = sqrt(1.69) c = 1.3 Step 8: Since 1.3 is already to the nearest tenth of a meter, no further rounding is needed. Final answer: 1.3 meters

8. √(13² + (2×6)²) = ? Answer: 17 Solution: Calculate 2×6 = 12 Square both numbers: 13² = 169 and 12² = 144 Add the squares: 169 + 144 = 313 Take the square root: √313 = 17 The answer is 17.
   Full step-by-step solution

   Step 1: Calculate 2×6 = 12 Step 2: Square both numbers: 13² = 169 and 12² = 144 Step 3: Add the squares: 169 + 144 = 313 Step 4: Take the square root: √313 = 17 The answer is 17.