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Published byQuentin Boise Modified about 1 year ago

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A general review Geometry

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

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

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

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30-60-90 Triangle (Pythagorean) c = 2a b = a√3 a 2a √3 a 30 60 90

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

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

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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

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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 2 ------------ (3-a) - a 9 – 6a + a 2 – a 2 ------------------ 3 – a – a == 9 – 6a ------- 3 – 2a = 3 -------- 3 – 2a =3 (slope)

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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

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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

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

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

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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

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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

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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 1 ---------- x 2 - x 1 change in y -------------------- change in x ∆y ----- ∆x y -- x

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

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

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

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Sum of all Interior Angles 180(n – 2) n is number of sides 180 360540

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Number of Diagonals in a Polygon n(n – 3) --------- 2 n is number of sides 0 25

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

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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)

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

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

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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

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

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

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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

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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

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

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

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

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

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Chords A line connecting two points of a circle

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

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Secant Lines Secant lines intersect two points of a curve

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

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