Pythagorean Theorem
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The document discusses the Pythagorean theorem, explaining that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples and exercises to find missing side lengths and to determine whether given triangles are right triangles based on their side lengths. Various figures and calculations illustrate the application of the theorem in different scenarios.
Pythagorean Theorem
- 3. c2 b2 a2 a2 +b2 = c2 b a c In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two legs. Hypotenuse Pythagorean Theorem
- 4. b a Cut the squares away from the right angle triangle and cut up the segments of square ‘a’ Draw line segment xy, parallel with the hypotenuse of the triangle x y q p Draw line segment pq, at right angles to Line segment xy. To show how this works:
- 5. Now rearrange them to look like this. You can see that they make a square with length of side ‘c’. This demonstrates that the areas of squares a and b add up to be the area of square c a2 +b2 = c2
- 6. Write an equation representing the relationship among the lengths of all sides of the following right triangle where the variables and numbers are units of length. 1) 𝑝 𝑞 𝑟 2) 8𝑛 𝑚 3) 𝑎 𝑏 13 𝑟2 + 𝑝2 = 𝑞2 82 + 𝑛2 = 𝑚2 𝑎2 + 𝑏2 = 132
- 7. Write an equation representing the relationship among the lengths of all sides of the following right triangle where the variables and numbers are units of length. 4) 𝑧 𝑥 𝑦 5) 6𝑒 14 7) 𝑡 𝑠 𝑟 8) 12 𝑎 7 6) 15 𝑏9 9) 16 𝑐 14 𝑥2 + 𝑦2 = 𝑧2 𝑒2 + 62 = 142 92 + 𝑏2 = 152 𝑠2 + 𝑡2 = 𝑟2 72 + 122 = 𝑎2 𝑐2 + 142 = 162
- 8. Find the lengths of the missing side of the right triangle in each of the following figures. 1) 20 21 3) 30 34 6.5 6 2) 212 + 202 = 𝑐2 441 + 400 = 𝑐2 841 = 𝑐2 𝑐 = 29 𝑎2 + 302 = 342 𝑎2 = 342 − 302 𝑎2 = 1156 − 900 𝑎2 = 256 𝑎 = 16 𝑎2 + 62 = 6.52 𝑎2 = 6.52 − 62 𝑎2 = 42.25 − 36 𝑎2 = 6.25 𝑎 = 2.5
- 9. Find the lengths of the missing side of the right triangle in each of the following figures. 4) 13 12 5) 11 21 6) 15 17 7) 12.5 3.5 8) 7.5 4.5 9) 37 35 5 8 5 8 12 6 12
- 10. The numbers in each of the following items are the lengths of the legs of the triangle. Find the length of the hypotenuse. 1) 6, 8 2) 7, 24 3) 18, 24 4) 4.5, 6 5) 2, 4.8 6) 1.5, 2 10 25 30 7.5 5.2 2.5
- 11. Pythagorean Theorem In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two legs. Reverse of the Pythagorean Theorem In a triangle, if the square of the longest side is equal to the sum of the squares of the other sides, then it is a right-angled triangle. Example: Let ∆𝐴𝐵𝐶 have 𝐴𝐵, 𝐴𝐶, and 𝐵𝐶, with 18, 24, and 30 centimeters long, respectively. Determine whether the ∆𝐴𝐵𝐶 is a right triangle. 𝐴 𝐵 𝐶 18 24 30 𝑎2 = 182 = 324 𝑏2 = 242 = 576 𝑐2 = 302 = 900 𝑐2 = 𝑎2 + 𝑏2 900 = 576 + 324 900 = 900
- 12. Example: Given ∆𝑃𝑄𝑅 where 𝑃𝑆 ⊥ 𝑄𝑅, and 𝑃𝑆, 𝑄𝑆, and 𝑆𝑅 have the lengths of 36, 27, and 48 units, respectively as shown in the figure. Show that ∆𝑃𝑄𝑅 is a right triangle. 𝑃 𝑄 𝑆 𝑅 36 27 48 𝑃𝑄2 = 362 + 272 𝑃𝑄2 = 1296 + 729 𝑃𝑄2 = 2025 𝑃𝑅2 = 362 + 482 𝑃𝑄2 = 1296 + 2304 𝑃𝑄2 = 3600 𝑃𝑄2 + 𝑃𝑅2 = 2025 + 3600 𝑃𝑄2 = 5625 𝑄𝑅 = 𝑄𝑆 + 𝑆𝑅 𝑄𝑅 = 27 + 48 𝑄𝑅 = 75 𝑄𝑅2 = 5625 𝑄𝑅2 = 𝑃𝑄2 + 𝑃𝑅2
- 13. Given the lengths of all the sides of triangles, determine whether each of the following triangles is a right triangle. 1) 14, 48, 50 2) 15, 24, 26 3) 24, 70, 74 4) 33, 56,65 5) 55, 47, 76 6) 10, 10.5, 14.5 7) 16.5, 90, 91.5 8) 4.2, 5.6, 8.4 9) 4.7, 5.5, 7.2 10) 7.5, 10, 12.5 right not right right right right right right not right not right right
- 14. Determine whether each of the following triangles is a right triangle. 1) 𝐴 𝐵 𝐶 30 12.5 72 2) 𝐴 𝐵 𝐶 30 32𝐷 24 right right
- 16. From the given figure, how many units is 𝑆𝑅 longer than 𝑄𝑆? 𝑃 𝑅 52 𝑄 𝑆 50 𝑃𝑄2 = 𝑃𝑆2 + 𝑄𝑆2 𝑄𝑆2 = 𝑃𝑄2 − 𝑃𝑆2 𝑄𝑆2 = 502 − 482 𝑄𝑆2 = 2500 − 2304 𝑄𝑆2 = 196 𝑄𝑆 = 1414 48 𝑃 𝑅 52 𝑆 48 𝑃𝑅2 = 𝑃𝑆2 + 𝑆𝑅2 𝑆𝑅2 = 𝑃𝑅2 − 𝑃𝑆2 𝑆𝑅2 = 522 − 482 𝑄𝑆2 = 2704 − 2304 𝑄𝑆2 = 400 𝑆𝑅 = 20 20 20 − 14 = 6 units 𝑄 𝑆 50 𝑃
- 17. 𝐷 𝐵 𝐶 𝐷 𝐴 𝐵 Given the figure of ∆𝐴𝐵𝐷 with an area of 24 square centimeters, if 𝐴𝐵 is 6 centimeters, how long is 𝐶𝐷? 𝐷 𝐴 𝐵 𝐶 6 Area of ∆𝐴𝐵𝐷 = 1 2 𝐴𝐵 𝐴𝐷 24 = 1 2 6 𝐴𝐷 𝐴𝐷 = 88 𝐵𝐷2 = 𝐴𝐷2 + 𝐴𝐵2 𝐵𝐷2 = 82 + 62 𝐵𝐷 = 10 10 10 𝐷 𝐵 𝐶 10 10 𝐶𝐷2 = 𝐵𝐷2 + 𝐵𝐶2 𝐶𝐷2 = 102 + 102 𝐶𝐷2 = 200 𝐶𝐷 = 10 2 10 2
- 18. Given that 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻 is a rectangular box as shown, the length of 𝐴𝐵 is 12 centimeters, 𝐵𝐶 is 9 centimeters, and 𝐴𝐹 is 8 centimeters. Find the length of 𝐴𝐻. 𝐴 𝐵 𝐶 12 9 𝐴𝐶2 = 𝐴𝐵2 + 𝐵𝐶2 𝐴𝐶2 = 122 + 92 𝐴𝐶 = 15 15 𝐻 𝐶 𝐴𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 𝐻 12 9 8 𝐺 15 8 𝐴𝐻2 = 𝐴𝐶2 + 𝐶𝐻2 𝐴𝐻2 = 152 + 82 𝐴𝐻2 = 289 𝐴𝐻 = 17 𝐴𝐶2 = 225 17
- 19. A ladder 13 meters in length rests with its top against the edge of the window. The foot of the ladder is placed 0.5 meters away from the building. How high is the edge of the window above the ground? 𝐴 𝐵 𝐶 13 𝑚 0.5 𝑚 𝐴𝐵2 = 𝐴𝐶2 − 𝐵𝐶2 𝐴𝐵2 = 132 − 1 2 2 𝐴𝐵2 = 169 − 1 4 𝐴𝐵2 = 675 4 𝐴𝐵 = 675 4 𝐴𝐵 = 15 3 2 15 3 2 𝑚
- 20. A ship travels to the north 20 miles, and to the west 2 miles before stopping. Then, it travels to the north 20 miles and moves to the east 11 miles. How many miles is this ship away from its origin? 𝐴 𝐵𝐶 𝐷 𝐸 20 2 20 11 𝐹 9 40 𝐴 𝐹 𝐸 40 9 𝐴𝐸2 = 𝐴𝐹2 + 𝐸𝐹2 = 92 + 402 = 81 + 1600 = 1681 𝐴𝐸 = 41 41
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