Math 10- Geometry
Chapter 1
I. Defining Terms
A. Undefined Terms
1. set- a group of objects w/ a common characteristic
2. point- a dot on paper
3. line- a set of points (endless, straight)
4. plane- a set of points forming a flat surface (endless)
B. Defined Terms
III. Definitions: Lines and Line Segments
A. Distance between two points
1. distance can never be negative
B. collinear set of points- a set of points all on the same line
C. segment- the set of all points between two endpoints
D. ray- the set of all points on part of a line including the dividing point
E. congruent- segments have the same length
F. midpoint- the point of a segment that divides it into two congruent segments
G. opposite rays- two rays that have the same endpoint and collinear points
IV. Angles
A. angle- the set of points representing the union of two rays
B. Interior/Exterior of an angle
V. Circles
A. circle- a set of points in a plane that are the same distance from the center
B. radius- a segment from the center of any point on the circle
C. arc- a part of a circle divided by 2 points on the circle
D. chord- a segment w/ 2 endpoints on the circle
E. diameter- a chord containing the center
F. semicircle- a part of a circle divided by the diameter
VI. Angles & their Measures
A. Right- 90
B. Straight- 180
C. Acute- 0 to 90
D. Obtuse- 90 to 180
E. Congruent angles have the same measure
VII. Pairs of Angles
A. adjacent angles- have a common vertex + side; no interior points in common
B. vertical angles- have a common vertex + their sides are 2 pairs of opposite rays
1. vertical angles are always congruent
C. complementary angles- angles that add up to 90
D. supplementary angles- angles that add up to 180
VIII. Definitions: Perpendicular Lines
A. perpendicular lines- lines that intersect and form right angles
B. perpendicular bisector- a line segment or ray that bisects + forms right angles w/ a segment
C. foot of a perpendicular- the point of intersection
D. distance from a point to a line- the length of the perpendicular
E. median- a segment joining any vortex to the midpoint of the opposite side
F. altitude- a segment drawn perpen. From a vertex to the opposite side
G. angle bisector- a segment that bisects an angle + ends in the opposite side
Chapter 2: Methods of Arriving at Conclusions
I. Using Observations
A. a proposition is a statement that is either true or false
B. Direct Observation- cant tell in geometry
II. Inductive Reasoning
A. exact measurements never exactly congruent
B. one counter-example is needed
III. Hypothesis & Conclusion
A. If, then sentence
1. If you practice, then you will succeed.
B. Simple Sentence
C.
2nd Trimester
Quadrilaterals
Parallelogram
Def: Quadrilateral where opposite sides are parallel
Properties:
opp. sides ||
opp. sides @
? opp. s @
? 2 consec. s suppl.
? diag. forms 2 @ D
diagonals bisect each other
To Prove:
if both pairs of opp. sides are @
? if both pairs of sides are ||
? if one side is || and @
if the diagonals bisect each other
? if both pairs of s are @
Rectangle
Def: A parallelogram w/ 1 rt.
Properties:
all of the properties of a parallelogram
? all 4 s are @
? diagonals are @
To Prove:
if its a ||ogram w/ a right
? if its a ||ogram w/ @ diagonals
Rhombus
Def: A parallelogram w/ 2 @ consec. sides
Properties:
all the properties of a (||ogram)
? the sides are all @
? the diagonals are ^ to each other
? diagonals bisect its s
To Prove:
if its a w/ 2 consec. sides @
? if its a whose diagonals are ^
? if its a w/ a diagonal bisecting the s
if its equilateral
Square
Def: A rectangle w/ 2 @ consecutive sides
Properties:
all the properties of a rectangle
? all the properties of a rhombus
? if a rhombus has 1 rt.
To Prove:
if its a rectangle w/ 2 consec. sides @
? if its a rhombus w/ at least 1 rt.
Trapezoid
Def: A quadrilateral that has 2 and only 2 sides ||
-isosceles trapezoid: a trapezoid where the nonparallel sides are @
-Properties:
the base s of an isos. trapezoid are @
? the diagonals are @
-To Prove:
if its a quadrilateral w/ only sides ||, and the nonparallel sides are @
Circles
Angles + Arcs
central angle- an whose vertex is the center of the circle
-equal to the arc
inscribed angle- an whose vertex is on the circle itself
-half the inscribed arc
minor arc- the set of points which lie in the interior of the central
major arc- the set of points which lie on the exterior of the central
-the measure of a minor arc is the samee as the central
-the measure of a major arc is 360 minnus the measure of the minor arc
-a quadrant is an arc that = 90
congruent arcs- arcs w/ = measures and = lengths
-if either the central , arc, or inscrribed chord is @, the other 2 are as well
-to prove arcs @, get chords or cent. s @ (all interchangeable)
-a diameter ^ to a chord bisects the chhord and its arcs
-if 2 chords are @, their distance fromm the center is =
Tangents + Secants
tangent- a line that intersects a circle at only 1 point
-its measure is half the difference of the intercepted arcs
-a tangent to a circle is ^ to a radius drawn to the point of contact
-if 2 lines of tangency are drawn from an external point, the segments are @
-the measure of an formed by a tangent and a chord = the intercepted
A circle is inscribed in a polygon if all sides are tangent to the circle.
secant- a line which intersects a circle in 2 points
-its measure is half the difference of the intercepted arcs
PGTh #1: all s outside are the difference of the 2 arcs
PGTh #2: all s inside are the sum of the arcs
Summary: to get arcs: find measure, cent. , inscribed , or formed by a tangent + chord
Line Segments and Similar Triangles
Ratio and Proportion
proportion- an equation which states that 2 ratios are =
-product of means = product of extremess
-in a proportion, if 3 terms are =, thee 4th will be too
When you have to find the 4th Proportional, you have to find d (a/b=c/d)
Mean Proportional: in a:b=b:c, b is the mean proportional
Similarity
Similar Polygons have @ s, and sides in proportion
-reflexive, symmetric, + transitive prooperties all apply to similarity
-to prove s @, show that theyre part of similar polygons
-to prove sides are proportion, do same thing (use corres. sides in proport.)
To Prove Triangles Similar:
-get 2 s @
-SAS, in proportion
-SSS, in proportion
-2 rt. Ds are similar if an acute of one is @ to an acute of the other
-a line that is || to 1 side of a D + iintersects the other sides cut off a D sim. to the given D
For problems w/ multiplication postulate, use the reason product of means = prod. of extremes
-in 2 sim. Ds, the corres. altitudes, pperimeters, diagonals, + line segments have same ratio
-sim. polygons may be divided into 2 siim. Ds
-in 2 similar polygons, the ratio of arrea is the ratio of the sides squared
-if 2 chords intersect in a circle, thee product of the 2 segments of the chords are @
-the distance from a point to a circle is the shortest distance to the edge
-secant A x its external segment = secant B x its external segment
-PA x CP = PB x DP
-tangent + secant tangent is mean prooportional b/w secant + its external segment
Projections
Projection of a point- the foot of the perpendicular drawn from a point to a line
Projection of a segment- the segment whose endpoints are projection
Right Triangles
If an altitude is drawn to the hypotenuse on a rt. ?:
the 2 ?s are similar to each other + the given ?
the alt. is the mean proportional b/w the lengths of the segments of the hyp.
Each leg is the mean proportional b/w the whole hyp. + the length of the projection of that leg on the hyp.
Pythagorean Theorem
-in a rt. ?, the square of the hyp. = tthe sum of the square of the legs
-triples: 3-4-5, 5-12-13, 8-15-17
-converse is also true
45-45 ?:
-legs are =
-the hyp. = leg x 2
-leg = hyp. x 2
-diagonal = side x 2
-side = diag. x 2
30-60 ?:
-leg opp. 30 = hyp.
-leg opp. 60 = the hyp. x 3
-longer = shorter leg x 3
-shorter leg = longer leg divided by 3
-shorter leg : hyp :: 1 : 2
-altitude = side 3
Areas of Polygons
Polygonal region- refers to the union of the polygon and its interior
Equivalent- having the same area
-if 2 ?s are @, they are equivalent
-if a polygon is divided, the sum of thhe parts equals the whole area
Rectangle- the product of its base and altitude
Triangle- base x height
-one side x its altitude = another side and its alt.
-rt. ?- product of 2 legs
-equiv. ?- (s2/4) x 3 (use when height is not given for equilateral triangle)
-?s w/ a common base + vertices that lie on a line II to the base are equivalent
Parallelogram- the product of any base and any corres. altitude
Square- side squared or the diagonal squared
Trapezoid- h(B +b)
-also altitude x median of 2 bases
Rhombus- the product of the diagonals
Equivalent Triangles
-when 2 ?s are @, they are also similarr and equivalent
-diagonal of a ||ogram creates 2 @, equuivalent ?s-2 ?s are equvalent if they have f bases + altitudes
-2 ?s are equivalent if they have a coommon base, and have vertices on a line || to the base-a median in a divides it into 2 equivalent s
-the ratio of the areas of 2 sim. ?s == the ratio of the square of the simila sides
*same for polygons
@ = ^ ~ D
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3rd Trimester
Testing Section 1
Chapter 8- Regular Polygons and the Circle
Definitions
regular polygons- equilateral + equiangular
circle circumscribed about a polygon- if circle passes through every vertex
-circles can only be circumscribed (+ inscribed in) about regular polygons
radius of a regular polygon- a radius of the circumscribed circle (that goes to the vertex)
central of a reg. polygon- an formed by 2 radii drawn to consecutive vertices of the polygon
has to equal 360/n [+ external] (w/ n as number of sides of the polygon)
inscribed circle- each side is tangent to the circle
apothem of a reg. polygon- a radius of its inscribed circle
center of a reg. polygon- the common center of the circumscribed + inscribed circles
Interior Angles
-a radius on a reg. polygon bisects thee of the polygon
Similarity
-all regular polygons w/ same # of sidees are similar
-radii, apothems, sides, + perimeters ((+ circumference) all in same ratio
-the area is the in proportion to the ssquare of the ratio
Area and Circumference
-area of a reg. polygon product of perimeter + apothem (A = ap)
-circumference a circles perimeter
-as the number of sides in a reg. polyggon increases, the perimeter approaches the
circumference of the circumscribed circle, but never gets there
-ratio of the circumference to the diammeter is 3.14:1 (22/7:1/:1)
-C = D
-length of an arc
-area of a circle r2 (+ the producct of its radius + circumference; d2)
-as the number of sides of a reg. polygon increases:
-the apothem approaches the radius
-area of reg. polygon approaches area of circle
-area of a sector
def: the union of an arc and 2 radii
-area of a segment area of minor secttor - area of D
def: the union of an arc, chord, + interior
Circles w/ Triangles
radius = 2x the apothem
radius = 2/3 the altitude
apothem = 1/3 the altitude
Testing Section 2
Ratios in Circles
circumference ratio equal to the ratio of the radii and diameter
area ratio equal to the square of the ratio of the radii and diameter
-same w/ area of sectors and segments
Chapter 10- Locus
Meaning of Locus
-only those points which satisfy a given condition; path
Discovering a Locus
-make a diagram w/ the given points
-decide the condition to be satisfied
-locate 1 point, then other points
-draw a line through these points
-describe the new figure
Fundamental Locus Theorems
-the locus of points equidistant from the ends of a segment is its perp. bisector
-the locus of points in the int. of an equidistant from the sides is its bisector
-the locus of points equidistant from 2 intersecting lines is the pair of lines which bisect
the s formed by these 2 lines
-the locus of points equidistant from 22 || lines is a 3rd || line midway b/w them
-the locus of points at a given distancce from a point is a circle w/ the given point as the
center, and the given distance as the radius
-the locus of points at a given distancce from a line is a pair of || lines
Intersections of Loci
-construct the locus of points that satisfy the 1st condition, then the 2nd
-determine the points of intersection of these loci
Chapter 11- Coordinate Geometry
Defs: abcissa x coordinate; ordinate y coordinate; origin 0,0
Distance pythagorean theorem (create a triangle); or d=(x2 - x1)2 = (y2 - y1)2
Midpoint averages of 2 pairs of coordinates
Slope change in y/change in x; vertical (1/0) no slope; horizontal (0/1) 0 slope
-if 2 lines have the same slope, theyre ||; if theyre negative reciprocals, theyre perp.