geometry review “say you’re me and you’re in math class….” geometry cohort weston middle...
TRANSCRIPT
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Geometry Review“Say you’re me and you’re in math class….”
Geometry CohortWeston Middle School
June 2013
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Definitions
• Point—0 Dimensions- P• Line Segment AB• Midpoint• Ray
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Definitions
• 2 Co-linear Points define a line• 2 intersecting lines define a plane
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Circle Definitions
• Circle• Radius• Chord- 2 points on a circle• Secant- line through 2 points• Diameter• Tangent• Major Arc, Minor Arc
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Angles
• 2 rays form angle• Vertex, Sides• Adjacent Angles-share a side• Acute Angle- < 90• Right Angles• Obtuse Angles> 90• Straight angle= 180• Reflex Angle > 180
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Angles-2
• Lines that intersect form Vertical angles• Vertical angles are equal• Supplementary angles add up to 180• Complementary angles add up to 90
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Parallel Lines and Transversals
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Triangle Types
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Triangles-1
• 3 points connected with line segments form triangle
• Sum of interior angles= 180 degrees• Exterior angle = sum of its remote interior
angles
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Congruent TrianglesTwo figures are congruent if they can sit on top of each other
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Congruent Triangles
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Perimeter, Area
• Perimeter= measurement around• Area
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Rectangle- 4 right angles, opposite sides equalRight Triangle- has 2 legs and a hypotenuseArea of right triangle= ½ product of legs
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Similarity
• Two figures are similar if one is simply a blown-up/rotated version of the other
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Similar Triangles
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Similar Triangles
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Right Triangles
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Pythagorean Formula
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Similar Trianglesb/d= a/e= (d+e)/b
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30-60-90 Triangle, Equilateral Triangle
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Similar Triangles
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Pythagorean Triples
• Set of three integers that satisfy the Pythagorean Theorem
Plimpton Tablet 1900 B.C.E.
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Heron’s Formula
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Angle Bisectors
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Angle Bisectors of a triangle are concurrent!
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Perpendicular Bisectors
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Euler Line
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Altitudes
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Medians
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Length of a median
• If the length of a median is ½ the length of the side to which it is drawn, the triangle must be a right triangle. Moreover, the side to which it is drawn is the hypotenuse
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Quadrilaterals- Interior angles = 360 degrees-
Alternative 1: Trapezoids have 1 pair of parallel sides
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Quadrilaterals- Alternative Arrangement- Trapezoids have at least one pair of parallel sides
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Trapezoids
• ( At least?)Two sides are parallel• Median= Average of the bases• Area= Height x median
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Parallelograms
• Both pairs of opposite sides equal• Opposite Angles Equal, consecutive angles
supplementary• Diagonals Bisect each other
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Rhombus
• All sides Equal• Diagonals Perpendicular• Area= half the product of diagonals
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KiteDefinition: A kite is a quadrilateral with two distinct pairs of
adjacent sides that are congruent.
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RectangleAll angles are 90 degrees
Area= l x wDiagonals= sqrt ( l^2 * w^2)
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Square
• All sides equal, all angles equal (90 degrees)• Diagonal= side*sqrt2
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Cyclic Quadrilateralseach exterior angle is equal to the opposite interior angle.
Sum of opposite angles= 180 degrees
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Brahmagupta’s Formula
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If and Only If
• Proving ‘If and Only If’ statements requires proving two different statements
“A month has less than thirty days if and only if the month is February”
To prove a statement true you must have a proof that covers all possibilities. To prove a statement false, you only have to show one counter-example.
Do not assume what you are trying to prove as part of a proof.
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Related Conditionals
• If the figure is a rhombus, then it is a parallelogram; converse is false
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Polygons
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Angles in a Polygon
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Area of a Regular Polygon
• Area= ½ x perimeter x apothem( distance from center to a side)
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Geometric Inequalities
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Geometric Inequalities
• When facing problems involving lengths of altitudes of a triangle, consider using area as a tool.
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Funky Areas
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Funky Areas
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Circles
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Circles
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Chords of a CircleIf chords of a circle are equal in length,
they subtend equal arcs
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Area of a Sector
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Thale’s Theorem
• Any Angle Inscribed in a semicircle is a right angle (Thale’s Theorem)
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Circle Angles
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Inscribed AnglesThe measure of an inscribed angle is ½ the arc it intercepts
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Inscribed AnglesAny two angles inscribed in the same arc are equal
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Intersecting Chords
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TangentsA Tangent Line is perpendicular to a radius
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Tangent Chords
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Secant-Secant Chords
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Secant-Secant Chords
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Two tangents are equal!
• Hat Rule:
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Power of a Point Theorem:Suppose a line through a point P intersects a circle in two points, U and V. For all such lines, the
product PU* PV is a constant.
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Power of a Point
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Planes
• Three non-collinear points determine a plane• Dihedral Angle: angle between the planes
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PrismsVolume = base x height
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3-D Figures
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Cube
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PyramidVolume = 1/3 *area of base*height
Lateral Surface Area= ½ Perimeter*slant height
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Cylinders
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ConesVolume = 1/3 * pi*r^2*h
LSA= pi*r*s
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Spheres
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Transformations
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TranslationsA translation "slides" an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face in the same direction. A
translation creates a figure that is congruent with the original figure and preserves distance (length) and orientation (lettering order). A translation is a direct isometry.
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Translations T (x,y)
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Rotations• A rotation is a transformation that turns a figure about a fixed point called
the center of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.(notation Rdegrees )
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Rotations R (x,y)
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Reflections• A reflection over a line k (notation rk) is a transformation in which each point of the
original figure (pre-image) has an image that is the same distance from the line of reflection as the original point but is on the opposite side of the line. Remember that a reflection is a flip. Under a reflection, the figure does not change size.
The line of reflection is the perpendicular bisector of the segment joining every point and its image.
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Reflections r(x,y)
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Glide ReflectionsReflection over a line + translation
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Dilations
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Dilations Dk(x,y)
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Compositions- Done in order from right to left
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Analytic Geometry
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Analytic Geometry
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Analytic GeometryThe product of slopes of perpendicular lines is -1
If two lines have the same slope, they are parallel
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Circle Equations
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Trigonometry
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Trigonometry
sin A= cos( 90-A)
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Law of Cosines
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Law of Sines
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Missing Lines
• The key to many problems is drawing the magic missing line
• Segments that stop inside figures should be extended
• Label lengths as you find them• If you see 30, 60, or 90 degree angles- buildm
30-60-90 degree triangles• When in doubt, build right triangles
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Good Luck!