4th year first term end exam reviewer math

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  • 8/4/2019 4th Year First Term End Exam Reviewer Math

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    4th Year First Term End Exam Reviewer

    Math

    I. Circles, Segments and Congruency- Circle

    o a group of points equidistant to a designated centero Divides its containing plane into 3 parts

    Area in the circle Area out of the circle Area on the circle itself

    o Written as (circle A) or (circle A with point x)o Secondary Parts

    Diameter chord that passes the center of the circle Radius- segment whose endpoints are a circles center and any point on the

    circle

    Chord a segment whose endpoints are points on a circle Tangent Line a line that makes contact with a circle at only 1 point Secant a line passing through a circle

    o Congruent vs. Concentric Concentric circles circles with the same center Congruent circles circles whose radii have the same length NOTE: You cant have a combination of the two. That would just be 1 circle

    - Sphereo Identical to a circle, except

    Its encased in a space 3 dimensional Has height, width, and depth

    - Inscriptiono When a circle overlaps all the vertices of a polygon, it either:

    Circumscribes about the polygon or the polygon is inscribed in the circle NOTE: Circles can also be inscribed in a polygon, Polygons can also circumscribe

    a circle

    Both are pretty much the same thing on paper if drawn. The only thing thatdiffers is perspective. One is from the inscribes point of view, the other about

    the circumscriber

    - Spheres and Planeso Null Set A sphere and plane that DO NOT intersecto Circle the intersection of a sphere and a planeo Great Circle the intersection of a sphere and a plane while the plane intersects the

    spheres center. Its the biggest intersection possible between the two

    - Special Triangles

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    o 30, 60, 90 30 = X 60 = X sqr. Root of 3 90 = 2X

    o 45, 45, 90 45 = x 90 = x sqr Root of 2

    - Common Tangentso A tangent common to 2 circles

    Common Internal

    Common External

    Internally Tangent

    Externally Tangent

    - Theorems in a nutshell

    II. Central Angles, Arcs, Chords- Central Angle

    o Angle whose vertex is the center of a circleo Create ARCS when intercepting a circle

    A segment of a circle 3 types

    Major > 180 deg Minor < 180 deg

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    Semi-Circle = 180 Measure of an Arc inside a central angle is equal to the measurement of the

    central angle

    - Inscribed Angleso Angles whose vertex is on a circle, and whose legs are chordso The arc inside the angle is twice the measure of the angle. Therefore.

    Inscribed angles on the same side of the circle with the same endpoints arecongruent

    Inscribed angles on opposite sides of the circle with the same endpoints aresupplementary

    - If things intersect and you want to get an angle.o Inside a circle

    Angle A = (arc BE +arc CD)(1/2)o On A circle

    Divide the intercepted arcby 2

    o Outside a circle In any of the 3 situations below, Subtract the smaller arc from the big arc, then

    divide by 2

    - If things intersect and you want to get a segment lengtho Of 2 chords

    (BA)(AC)=(EA)(AD)

    o Of 2 Secants (AB)(AC)=(AD)(AE)

    o Of a Secant and Tangent (AB)=(AC)(AD

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    III. Conic Sections- Sections generated by cones and a plane- 4 types, but for now well use only 3

    1) Circle- Already talked about this one, so lets skip ahead- Equations

    o Standard r = x + y

    thats if your origin is 0 r = (x-h) + (y-k)

    thats if your origin is (h,k)o General

    x - 2xh + h + y - 2yk + k = r- If given 3 points and asked to find the circle

    o Make a general equation for each of the points, substituting X and Y from the equationusing the coordinates

    o Through Elimination, isolate a value (youre choice here)o When you get your value, use it to solve for the remaining value

    2) SemiCircle- Half a circle- Equations

    o If semi-circle is either the TOP or BOTTOM half of the circle and its center is the origin TOP: y = r - x BOTTOM: y = - (r - x)

    o If semi-circle is either the LEFT or RIGHT half of the circle and its center is the origin RIGHT: x = r - y LEFT: x = -(r - y)

    o If your center isnt the origin, substitute x in any of the equations with (x-h) and y with(y-k), your vertex being (h,k)

    3) Parabola- A locus of points equidistant from a Directrix and a Focus- Parts:

    o Vertex: midpoint between focus and vertexo Axis of Symmetery: line that passes through the

    Focus and Vertex. Indicates Whether Vertical or

    Horizontal Parabola

    o Focal Width: Distance Between Vertex andFocus, Vertex and Directrix

    o Latus Rectum: Passes through the focus and its endpoints are on the parabola. Length is4p

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    o Directrix:Important part, but we really dont use it a lot in tests. Is P away from thevertex, and 2p away from the focus

    o Focus: needed to graph pretty much everything. Is P away from the vertex, and 2p awayfrom the directrix

    - Equationo If parabola is either pointing UP or DOWN and its vertex is the origin

    UP: x = 4py DOWN: x = - 4py

    o If parabola is pointing either LEFT or RIGHT and its vertex is the origin RIGHT: y = 4px LEFT: y = - 4px

    o If your vertex isnt the origin, substitute x in any of the equations with (x-h) and y with(y-k) if your vertex is (h,k)