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A general review Geometry. Pythagorean Theorem a 2 + b 2 = c 2 Examples: 3, 4 = 5 5, 12 = 13 8, 15 = 17 a b c.

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Presentation on theme: "A general review Geometry. Pythagorean Theorem a 2 + b 2 = c 2 Examples: 3, 4 = 5 5, 12 = 13 8, 15 = 17 a b c."— Presentation transcript:

1 A general review Geometry

2 Pythagorean Theorem a 2 + b 2 = c 2 Examples: 3, 4 = 5 5, 12 = 13 8, 15 = 17 a b c

3 Three Dimensions X, Y, Z axis (x, y, z) z x y

4 Length of a line in 3D x 2 + y 2 + z 2 = length 2 x y z length

5 Triangle (Pythagorean) c = a√2 a √2 a a 45 90

6 Triangle (Pythagorean) c = 2a b = a√3 a 2a √3 a

7 Midpoint x avg = (x 1 + x 2 )/2 y avg = (y 1 + y 2 )/2 (x1, y1) (x2, y2) (, ) x 1 + x y 1 + y

8 Slope Formula The slope from (x 1, y 1 ) to (x 2, y 2 ): y 2 - y x 2 - x 1 (x1, y1) (x2, y2) y 2 - y 1 x 2 - x 1

9 Collinear Points Points are collinear when they all have the same slope --or-- Points are collinear when they all fall on the same line Collinear Non-Collinear

10 Random Fact! If the x coordinates between two points on the graph y = x 2 add up to 3, then the slope between those two points equals 3. Let’s say: P1 = (a, a 2 ) and P2 = (3-a, (3-a) 2 ) (3-a) 2 – a (3-a) - a 9 – 6a + a 2 – a – a – a == 9 – 6a – 2a = – 2a =3 (slope)

11 Distance Formula Using Pythagorean theorem: (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 = length 2 x 2 - x 1 y 2 - y 1 length

12 Distance Formula in 3D (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 + (z 2 – z 1 ) 2 = length 2 x 2 – x 1 y 2 – y 1 z 2 – z 1 length

13 Circle Formula At midpoint (a, b): (x – a) 2 + (y – b) 2 = r 2 r (a, b)

14 Sphere Formula At midpoint (a, b, c): (x – a) 2 + (y – b) 2 + (z – c) 2 = r 2 (a, b, c) r

15 Parallel and Perpendicular Parallel lines will have the same slope Perpendicular lines: if one line has slope m the other line has slope -1/m Slope: m Slope: -1/m

16 Movement Movement in 2D is symbolized by where a is the change in x where b is the change in y Movement in 3D is symbolized by where a is the change in x where b is the change in y where c is the change in z

17 Slope (fun facts) Since slope is: Slope can also be: ∆ means “change in”, so slope can also be: In terms of movement, the slope is simply: y 2 - y x 2 - x 1 change in y change in x ∆y ∆x y -- x

18 Perpendicular Movements in 3D Two movements are perpendicular if: ax + by + cz = 0 Movement 1 is Movement 2 is

19 Movements Perpendicular to Plane Ax + By + Cz = D A, B, C, D are constant is always perpendicular to is only moving on one plane

20 Convex / Concave Polygons Convex: none of the angles > 180 Concave: 1 or more angles > 180 Convex PolygonConcave Polygon

21 Sum of all Interior Angles 180(n – 2) n is number of sides

22 Number of Diagonals in a Polygon n(n – 3) n is number of sides 0 25

23 Regular Polygon All sides and angles are the same 60 aa a

24 Perimeter and Area Perimeter of Circle: 2 π r Perimeter of Polygon: Sum of all sides Area of Circle: π r 2 Area of Rectangle: width x height Area of Triangle: ½ base x height Area of Trapezoid: ½ (base 1 + base 2 ) x height Area of a Regular Polygon: n(s 2 /4)tan((n-2)180/2n) Area of a Cone’s side: π r (height of slant)

25 Prism Surface Area Area = (2 * base area) + (base perimeter * height) r h Area = 2 π r π rh

26 Prism Volume Volume = base area * height r h Volume = π r 2 h

27 Pyramid/Cone Surface Area Area = base area + sum of side area(s) ∑ means “sum of” l 2 = r 2 + h 2 Area = π r 2 + π rl h r l

28 Pyramid/Cone Volume Volume = (1/3) base area * height Volume = (1/3) π r 2 h h r

29 Sphere Surface Area and Volume Area = 4 π r 2 Volume = 4/3 π r 3 r

30 Similar Figures Have equal angles and a constant ratio between their sides R = Ratio between their side lengths R = Ratio between their perimeters R 2 = Ratio between their areas Side ratio: 2:1 Perimeter ratio: 2:1 Area ratio: 4:1

31 Similar Figures in 3D R = Ratio between their side lengths R = Ratio between their perimeters R 2 = Ratio between their areas R 3 = Ratio between their volumes Side ratio: 2:1 Perimeter ratio: 2:1 Area ratio: 4:1 Volume ratio: 8:1

32 Inscribed Angle Angle that touches side Arc Angle – perimeter which inscribed angle intersects Inscribed Angle Arc Angle

33 Inscribed Angle The measure of an inscribed angle that touches the side is: Arc Length/2 Arc Length: π Inscribed Angle: π /2

34 Tangent Lines A tangent line hits only one point of a curve Point of tangency

35 Tangent Lines Imply Equal Lengths Two intersecting tangent lines have equal lengths from their points of tangency.

36 Chords A line connecting two points of a circle

37 Intersecting chords AB * BD = AC * CD A B C D

38 Secant Lines Secant lines intersect two points of a curve

39 Intersecting Secant Lines = A B D C E AB AC AE AD


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