geometry finals (semester 2) study guide :)
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VOCAB
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plane - a surface such that if any two points on the surface are connected by a line, all points of the line are also on the surface.
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noncoplanar - not on the same plane.
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coplanar - on the same plane.
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foot - point of intersection of a line and a plane.
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POSTULATES
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Three noncollinear points determine a plane.
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If a line intersects a plane not containing it, then the intersection is exactly one point.
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If two planes intersect, their intersection is exactly one line.
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THEOREMS
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A line and a point not on the line determine a plane.
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Two intersecting lines determine a plane.
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Two parallel lines determine a plane.
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If a line is perpendicular to 2 distinct lines that lie on a plane and that pass through its foot, then it is perpendicular to the plane.
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If a plane intersects 2 parallel planes, the lines of intersection are parallel.
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THEOREMS
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Means-Extremes Products Theorem - a/b = c/d -> ad=bc
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Means-Extremes Ratio Theorem - pq=rs -> p/r=s/q etc.
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Arithmetic mean example: given 3 & 7, 3+27/2= 15
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Geometric mean example: given 3 & 7, 3/x = x/27 x=+ or - 9
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AAA (angles) - similar
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AA (angles) - similar
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Side-Splitter Theorem - ab/bc = ae/ed
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TERMS
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Dilation/Reduction
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Similarity - same shape but not size
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TERMS
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concentric circle - same center with different size
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chord - points connected by a segment on a circle
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diameter/radius
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secants/tangents
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THEOREMS
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If a radius is perpendicular to a chord then it bisects it (reversed too).
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The perpendicular bisector of a chord passes through the center of a circle.
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If 2 chords are equidistant from the center then they are congruent (reversed too).
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secant/tangent theorems - example: 1/2 (large angle - medium angle) = small angle
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chords: ev . en = el . se
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tp (tangent) squared = (pr)(pq aka external part of secant)
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pb . pa (external part of secant) = pd . pc (external part of secant)
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TERMS
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bases
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lateral faces
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lateral edges
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slant height
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altitude (height)
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LA: Lateral surface area - no bases
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TA: Total surface area - with bases
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volume
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THEOREMS
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The sum of the measures of the 3 angles of a triangle is 180.
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The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
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A segment joining the midpoints of 2 sides of a triangle is parallel to the third side, and its length is one-half the length of the third side (midline theorem).
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No choice theorem - if 2 angles of a triangle are congruent then the remaining ones are also.
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AAS - angle angle side
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FORMULAS
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sum of angles in polygon = (sides - 2)180
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exterior angles = 360
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diagonals = sides(sides - 3)/2
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exterior angle = 360/sides
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RADICAL REVIEW
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squared root of 48 = 4 radical 3
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squared root of 5/3 = squared root of 15/3
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CIRCLES
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circumference - pi d
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area - pi r squared
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sector - fraction of circle area
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arc - fraction of circumference
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secants - through circle
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tangent - edge of circle (external/internal)
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RIGHT TRIANGLE ALTITUDES
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h squared = x . y
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a squared = x . c
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b squared = y . c
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OTHER
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pythagorean theorem - a squared + b squared = c squared
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distance formula - squared root (x2 - x1) squared + (y2 - y1) squared
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30 60 90
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45 45 90
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SOH CAH TOA
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I don't feel like writing all of the area formulas but here are the ones you need to know...
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square/rectanle
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triangle
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parallelogram
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trapezoid (and median)
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kite
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polygons
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circle, sectors, segments
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hero formula: squared root of s(s-a)(s-b)(s-c)
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