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Geometry A general review

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**Pythagorean Theorem a2 + b2 = c2 Examples: 3, 4 = 5 5, 12 = 13**

8, 15 = 17 c a b

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

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Length of a line in 3D x2 + y2 + z2 = length2 length z y x

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**45-45-90 Triangle (Pythagorean)**

c = a√2 45 √2 a a 90 45 a

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**30-60-90 Triangle (Pythagorean)**

c = 2a b = a√3 30 2a √3 a 90 60 a

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**Midpoint xavg = (x1 + x2)/2 yavg = (y1 + y2)/2 (x2, y2) x1 + x2**

2 y1 + y2 2 ( , ) (x1, y1)

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**Slope Formula The slope from (x1, y1) to (x2, y2): y2 - y1 ----------**

x2 - x1 (x2, y2) y2 - y1 (x1, y1) x2 - x1

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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 = x2 add up to 3, then the slope between those two points equals 3. Let’s say: P1 = (a, a2) and P2 = (3-a, (3-a)2) (3-a)2 – a2 (3-a) - a 9 – 6a + a2 – a2 3 – a – a 9 – 6a 3 – 2a 3 – 2a = = = = 3 (slope)

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**Distance Formula Using Pythagorean theorem:**

(x2 – x1)2 + (y2 – y1)2 = length2 length y2 - y1 x2 - x1

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**Distance Formula in 3D (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2 = length2**

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Circle Formula At midpoint (a, b): (x – a)2 + (y – b)2 = r2 (a, b) r

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**Sphere Formula At midpoint (a, b, c):**

(x – a)2 + (y – b)2 + (z – c)2 = r2 (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: -1/m Slope: m

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**Movement Movement in 2D is symbolized by <a, b>**

where a is the change in x where b is the change in y Movement in 3D is symbolized by <a, b, c> 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 <x, y>, the slope is simply: y2 - y1 x2 - x1 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 <a, b, c> Movement 2 is <x, y, z> <a, b, c> <x, y, z>

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**Movements Perpendicular to Plane**

Ax + By + Cz = D A, B, C, D are constant <x, y, z> is always perpendicular to <A, B, C> <x, y, z> is only moving on one plane <A, B, C> <x, y, z>

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**Convex / Concave Polygons**

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

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**Sum of all Interior Angles**

n is number of sides 360 540 180

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**Number of Diagonals in a Polygon**

n(n – 3) 2 n is number of sides 2 5

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

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**Perimeter and Area Perimeter of Circle: 2πr**

Perimeter of Polygon: Sum of all sides Area of Circle: πr2 Area of Rectangle: width x height Area of Triangle: ½ base x height Area of Trapezoid: ½ (base1 + base2) x height Area of a Regular Polygon: n(s2/4)tan((n-2)180/2n) Area of a Cone’s side: πr (height of slant)

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**Prism Surface Area Area = (2 * base area) + (base perimeter * height)**

Area = 2πr 2+ 2πrh h

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

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**Pyramid/Cone Surface Area**

Area = base area + sum of side area(s) ∑ means “sum of” l2 = r2 + h2 Area = πr 2+ πrl l h r

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**Pyramid/Cone Volume Volume = (1/3) base area * height**

Volume = (1/3) πr 2h h r

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**Sphere Surface Area and Volume**

Area = 4πr2 Volume = 4/3 πr3 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 R2 = 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 R2 = Ratio between their areas R3 = 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 Arc Angle Inscribed 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 D A B C

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Secant Lines Secant lines intersect two points of a curve

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**Intersecting Secant Lines**

= AB AC AE AD D B E A C

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