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Pythagorean Theorem Explained: Find Areas and Diagonal Distances Easily

Learn how to apply the Pythagorean Theorem to find areas, side lengths, and diagonal distances in squares and triangles. Discover shortcuts and practical applications to simplify calculations in geometry lessons.

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Pythagorean Theorem Explained: Find Areas and Diagonal Distances Easily

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  1. 8th Grade Math 1st Period Nov. 1, 2012

  2. No HW Review; Start Warm-Up Find the areas of the following: 1. 2. 3. 4. 5.

  3. Warm-Up: Answers Find the areas of the following: 1. 2. 3. 4. 5. What operation could be applied to the area in #5 to get the side length of the square? A = 2 square units A = 1 square unit A = 4 square units A = 5 square units A = 1 square unit

  4. So, to find the diagonal distance between two points, you can … • Draw a line segment between the points. • Make a square with the segment as a side. • Find the area of the square. • Take the square root of the area.

  5. But … Do We Have to Make the Square Each Time?

  6. No! Notice that … = making + =

  7. If we remember our original line segment … is the hypotenuse of and we call the side lengths a, b, and c … a2 c b a + = b2 c2

  8. That is, we can find … the square of the hypotenuse by adding the squares of the legs c2 b2 a2 This fact is known as the Pythagorean Theorem!

  9. Looking for length of the hypotenuse a2 + b2 = c2 152 + 202 = x2 225 + 400 = x2 625 = x2 25 = x Using the Pythagorean Theorem x 15 20

  10. Looking for length of a leg a2 + b2 = c2 62 + x2 = 102 36 + x2 = 100 -36-36 x2 = 64 x = 8 Using the Pythagorean Theorem 10 6 x

  11. Looking for the diagonal distance a2 + b2 = c2 32 + 42 = x2 9 + 16 = x2 25 = x2 5 = x Using the Pythagorean Theorem x 3 4

  12. HW (Front & Back of the Same WS) • Applications #1-2, 5-6 (circled) • Using the Pythagorean Theorem in Word Problems #1 (circled)

  13. 8th Grade Math 2nd Period Nov. 1, 2012

  14. No HW Review; Start Warm-Up Find the areas of the following: 1. 2. 3. 4. 5.

  15. Warm-Up: Answers Find the areas of the following: 1. 2. 3. 4. 5. What operation could be applied to the area in #5 to get the side length of the square? A = 2 square units A = 1 square unit A = 4 square units A = 5 square units A = 1 square unit

  16. So, to find the diagonal distance between two points, you can … • Draw a line segment between the points. • Make a square with the segment as a side. • Find the area of the square. • Take the square root of the area.

  17. But … Do We Have to Make the Square Each Time?

  18. No! Notice that … = making + =

  19. If we remember our original line segment … is the hypotenuse of and we call the side lengths a, b, and c … a2 c b a + = b2 c2

  20. That is, we can find … the square of the hypotenuse by adding the squares of the legs c2 b2 a2 This fact is known as the Pythagorean Theorem!

  21. Looking for length of the hypotenuse a2 + b2 = c2 152 + 202 = x2 225 + 400 = x2 625 = x2 25 = x Using the Pythagorean Theorem x 15 20

  22. HW (Front & Back of the Same WS) • Applications #1-2, 5-6 (circled) • Using the Pythagorean Theorem in Word Problems #1 (circled)

  23. 8th Grade Math 4th Period Nov. 1, 2012

  24. No HW Review; Start Warm-Up Find the areas of the following: 1. 2. 3. 4. 5.

  25. Warm-Up: Answers Find the areas of the following: 1. 2. 3. 4. 5. What operation could be applied to the area in #5 to get the side length of the square? A = 2 square units A = 1 square unit A = 4 square units A = 5 square units A = 1 square unit

  26. So, to find the diagonal distance between two points, you can … • Draw a line segment between the points. • Make a square with the segment as a side. • Find the area of the square. • Take the square root of the area.

  27. But … Do We Have to Make the Square Each Time?

  28. No! Notice that … = making + =

  29. If we remember our original line segment … is the hypotenuse of and we call the side lengths a, b, and c … a2 c b a + = b2 c2

  30. That is, we can find … the square of the hypotenuse by adding the squares of the legs c2 b2 a2 This fact is known as the Pythagorean Theorem!

  31. Looking for length of the hypotenuse a2 + b2 = c2 152 + 202 = x2 225 + 400 = x2 625 = x2 25 = x Using the Pythagorean Theorem x 15 20

  32. Looking for the diagonal distance a2 + b2 = c2 32 + 42 = x2 9 + 16 = x2 25 = x2 5 = x Using the Pythagorean Theorem x 3 4

  33. HW (Front & Back of the Same WS) • Applications #1-2, 5-6 (circled) • Using the Pythagorean Theorem in Word Problems #1 (circled)

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