Pythagorean Theorem and 3-D Figures

1. Pythagorean Theorem and 3-D Figures

2. Rectangular solids • Every rectangular solid has: • 6 rectangular faces: • 1. • 2. • 3. • 4. • 5. • 6.

3. 12 edges: • 1. 2. • 3. 4. • 5. 6. • 7. 8. • 9. 10. • 11. 12.

4. 4 diagonals: • 1. • 2. • 3. • 4. • A cube is a rectangular solid in which all edges are congruent. All faces are squares!

5. Regular Square Pyramid • KLQP is a square and is called the base. • J is the vertex. • is the altitude of the pyramid and is perpendicular to the base at its center. • is called the slant height and is perpendicular to a side of the base.

6. Example 1: This box is a rectangular solid. • If BC = 3, what is AD? • If AB = 4 what is DB? • Must use BC = 3 to find • If CG =12, what is DH? • What is BH? AG? • Use DB = 5

7. Example 2: • The dimensions of a rectangular solid are 3, 5, and 7. Find the diagonal. • Draw a rectangular solid. • Does it matter where the numbers go? • 1st possibility • Find AH • To find AH first must find DH • Use DH to find AH

8. Example 2: • 2nd possibility • Find AH • To find AH first must find DH • Use DH to find AH

9. Example 2: • 3rd possibility • Find AH • To find AH first must find DH • Use DH to find AH

10. Example 3: • The square pyramid has altitude , slant height , perimeter of JKMO = 40, and PK = 13. • Find • a.) JK • Since it is a square pyramid, • the base is a square • A square has 4 congruent sides • If the perimeter is 40 then each • side is 10 so JK = 10

11. b.) PS • In a square pyramid, the slant • height bisects the base edge • So KS = 5 • Use PK = 13 to find PS • PS = 12

12. C.) PR • In a square pyramid, the altitude • height bisects the base edge • So RS = 5 • Use PS = 12 to find PR

13. In review, • With any type of solid, you need to create right triangles that you can use the Pythagorean to find the missing length. • You may need to use the Pythagorean theorem more than once to find the missing length.