Download File

Christina Yee
Advanced Geometry Section 5
Christina’s Guide to Similar Polygons, Circles, and Surface Area
and Volume
CHAPTER 8: SIMILAR POLYGONS
8.1: Ratio and Proportion
Ratio: The quotient of two numbers
𝑎
Proportion: An equation stating that two or more ratios are equal. Example: 𝑏 =
𝑐
𝑑
or a:b = c: d
a = first term (extreme), b = second term (mean), c = third term (extreme), d = fourth term (mean)
Product and Ratio Theorems
Theorem 59: In a proportion, the product of the means is equal to the product of the extremes. (Means-Extremes Products
Theorem.)
𝑎
𝑐
Ex: If = then ad = bc
𝑏
𝑑
Theorem 60: If the Product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then
either pair of numbers may be made the extremes, and the other pair the means, of a proportion. (Means-Extremes Ratio
Theoroem.)
Ex: If ab = cd, then
𝑎
𝑑
𝑐 𝑎
𝑏 𝑐
= ,
𝑑
𝑏
= , and
𝑑
𝑎
=
𝑏
𝑐
Mean proportion: If the means in a proportion are equal, either mean is called a geometric mean, or mean proportional,
between the extremes.
2
Ex: 6 =
6
18
(Here, both of the means are 6) This means 6 is the geometric mean between 2 and 18.
Arithmetic mean: The average of two numbers
Ex: (2 + 18)/2 = 10. 10 is the average of 2 and 18.
8.2 Similarity
Similar polygons: Polygons in which (1) the ratios of the measures of corresponding sides are equal and (2) corresponding
angles are congruent.
Polygon 1
Polygon 2
5
In the diagram above, 10 =
polygon 2.
3
6
1
and 2 =
2
4
5
1
and 10 = 2. Thus, both polygons are similar. This can be written as polygon 1 ~
Theorem 61: The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides. Ex:
5+2+1+3
10+4+2+6
=
3
6
8.3 Methods of Proving Triangles Similar
Postulate: If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle
are congruent to the corresponding angles of the other triangle, then the triangles are similar. (AAA)
Theorem 62: If there exists a correspondence between the vertices of two triangles such that two angles of one triangle are
congruent to the corresponding angles of the other, then the triangles are similar. (AA)
Theorem 63: If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of
corresponding sides are equal, then the triangles are similar. (SSS~)
Theorem 64: If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of
two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar. (SAS~)
8.4 Congruencies and Proportions in Similar Triangles
Once we prove that two triangles are similar, we can use the definition of similar polygons to prove:
1.) Corresponding sides of the triangle are proportional. (The ratios of the measures of the corresponding sides are equal.)
2.) Corresponding angles of the triangle are congruent.
8.5 Three Theorems Involving Proportions
Theorem 65: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides
proportionally.
Theorem 66: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals
proportionally.
Theorem 67: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the
adjacent sides. (Angle Bisector Theorem)
CHAPTER 10: THE CIRCLE
10.1 The Circle
Circle: The set of all points in a plane that are a given distance from a given point in the plane. The given point is the
center of the circle, and the given distance is the radius. A segment that joins the center to a point on the circle is also
called a radius.
Cocentric circles: two or more coplanar circles with the same center.
Congruent circles: when two circles have congruent radii.
Point in the interior of a circle: when a point’s distance from the center is less than the radius.
Point in the exterior of a circle: when a point’s distance from the center is greater than the radius.
Point on the circle: when a point’s distance from the center is equal to the radius.
Chord: segment joining any two points on the circle
Diameter: a chord that passes through the center of the circle.
Radius – chord relationships
Distance: the distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to
the chord.
distance = 6 cm
Theorem 74: If a radius is perpendicular to a chord, then it bisects the chord.
Theorem 75: If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord.
Theorem 76: The perpendicular bisector of a chord passes through the center of the circle.
10.2 Congruent Chords
Theorem 77: If two chords of a circle are equidistant from the center, then they are congruent.
Theorem 78: If two chords of a circle are congruent, then they are equidistant from the center of the circle.
If AE = AF, then BG is congruent to DC. If DC is congruent to BG, AE = AF.
10.3 Arcs of a Circle
Arc: two points on a circle and all points on the circle needed to connect the points by a single path.
Center of an arc: center of the circle of which the arc is a part.
Central angle: angle whose vertex is at the center of a circle.
Minor arc: arc whose points are on or between the sides of a central angle.
Major arc: arc whose points are on or outside of a central angle.
Semicirlce: an arc whose endpoints are endpoints of a diameter.
Measures of Arcs
The measure of a minor arc or a semicircle is the same as the measure of the central angle that intercepts the arc. The
measure of a major arc is 360 minus the measure of the minor arc with the same end points.
Congruent Arcs
Two arcs are congruent when they have the same measure and are parts of the same circle or congruent circles.
Arc AD is congruent to arc BC.
Theorems relating arc, chords, and central angles
Theorem 79: If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are
congruent.
Theorem 80: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are
congruent.
Theorem 81: If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are
congruent.
Theorem 82: If two chords of a circle (or of congruent circles) are congruent, then the corresponding central angles are
congruent.
Theorem 83: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.
Theorem 84: If two chords of a circle (or of congruent circles) are congruent, then the corresponding arcs are congruent.
10.4 Secants and Tangents
Secant: a line that intersects a circle at exactly two points. (Every secant contains a chord of the circle.
Tangent: a line that intersects a circle at exactly one point. This point is called the point of tangency or point of contact.
Postulates:
1.) A tangent is perpendicular to the radius drawn to the point of contact.
2.) If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle.
Tangent segment: part of a tangent line between the point of contact and a point outside the circle.
Secant segment: part of a secant line that joins a point outside the circle to the farther intersection point of the secant and
the circle.
External part: the part of a secant line that joins the outside point to the nearer intersection point.
PR is a secant segment.
Theorem 85: I two tangent segments are drawn to a circle from an external point, then those segments are congruent.
(Two tangent theorem.)
Tangent circles
Tangent circles: circles that intersect each other at exactly one point.
Externally tangent: when each of the tangent circles lies outside the other.
Internally tangent: when one of the tangent circles lies inside the other.
Common tangents: A common tangent is a line tangent to two circles (not necessarily at the same point.) Such a tangent
is a common internal tangent if it lies between the circles (intersects the segment joining the centers AKA the line of
centers) or a common external tangent if it is not between the circles (does not intersect the segment joining the
centers).
AB is a common internal tangent, CD is a common external tangent.
How to solve common-tangent problems:
1.) Draw the segment joining the centers.
2.) Draw radii to the points of contact.
3.) Through the center of the smaller circle, draw a line parallel to the common tangent.
4.) Observe that this line will intersect the radius of the larger circle to form a rectangle and a right triangle.
5.) Use the Pythagorean theorem an properties of a rectangle.
10.5 Angles Related to a Circle
Inscribed angle: an angle whose vertex is on a circle and whose sides are determined by two chords.
Tangent-chord angle: and angle whose vertex is on a circle and whose sides are determined by a tangent and a chord that
intersect at the tangent’s point of contact.
Theorem 86: The measure of an inscribed angle or a tangent-chord angle (vertex on a circle) is one-half the measure of its
intercepted arc.
Angles with vertices inside, but not at the center of, a circle
Chord-chord angle: an angle formed by two chord that intersect inside a circle but not at the center.
Theorem 87: The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chordchord angle and its vertical angle.
Angles with vertices outside a circle
Secant-secant angle: an angle whose vertex is outside a circle and whose sides are determined by two secants
Secant-tangent angle: an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent.
Tangent-tangent angle: and angle whose vertex is outside a circle and whose sides are determined by two tangents.
Theorem 88: The measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle (vertex outside a
circle) is on-half the difference of the measures of the intercepted arcs.
Secant-secant angle
secant-tangent angle
Tangent-tangent angle
10.6 More Angle-Arc Theorems
Congruent inscribed and tangent-chord angles
Theorem 89: If two inscribed or tangent-chord angles intercept the same arc, then they are congruent
Theorem 90: If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruent.
Angles inscribed in semicircles
Theorem 91: An angle inscribed in a semicircle is a right angle.
A special theorem about tangent-tangent angles
Theorem 92: The sum of the measures of a tangent-tangent angle and its minor arc is 180.
10.7 Inscribed and Circumscribed Polygons
Inscribed in: when a polygon’s vertices all lie on the circle.
Circumscribed about: when a polygon’s sides are all tangent to the circle.
Circumcenter: the center of a circle circumscribed about a polygon.
Incenter: the center of a circle inscribed in a polygon.
A theorem about inscribed quadrilaterals
Theorem 93: If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.
Theorem 94: If a parallelogram is inscribed in a circle, it must be a rectangle.
10.8 The Power Theorems
Theorem 95: If two chords of a circle intersect inside the circle, then the product of the measures of the segments of one
chord is equal to the product of the measures of the segments of the other chord. (Chord-chord power theorem)
Theorem 96: If a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the
measure of the tangent segment is equal to the product of the measures of the entire secant segment and its external part.
(Tangent-secant power theorem.)
Theorem 97: If two secant segments are drawn from an external point to a circle, then the product of the measures of one
secant segment and its external part is equal to the product of the measures of the other secant segment and its external
part. (Secant-Secant power theorem.)
10.9 Circumference and Arc Length
Circumference: the perimeter of a circle. C = πd
Theorem 98: The length of an arc is equal to the circumference of its circle times the fractional part of the circle
determined by the arc.
Length of arc PQ =
Where d is the diameter and arc PQ is measured in degrees.
𝑚 𝑜𝑓 𝑎𝑟𝑐𝑃𝑄
360
* πd
CHAPTER 12: SURFACE AREA AND VOLUME
12.1 Surface Areas of Prisms
Polyhedra: solids with flat faces.
Prism: a polyhedral with two congruent parallel faces and a set of parallel edges that connect corresponding vertices of the
two parallel faces.
Lateral surface area: the sum of the areas of the lateral faces.
Total surface area: the sum of the prism’s lateral area and the areas of the two bases.
In prisms, if the lateral edges are perpendicular to the bases, then the lateral faces will be rectangles.
SA = LA + 2B where B = the area of the base
12.2 Surface Areas of Pyramids
Pyramid: has only one base and its lateral edges are not parallel but meet at a single point called the vertex. The base may
be any type of polygon, but the lateral faces are always triangles.
Regular pyramid: has a regular polygon as its base and has congruent lateral edges.
height = altitude
SA = LA + B
Volume = B*height*1/3
12.3 Surface Areas of Circular Solids
Cylinder: resembles a prism with two circular bases.
Theorem 113: The Lateral area of a cylinder is equal to the product of the height and the circumference of the base.
𝐿𝐴𝑐𝑦𝑙 = Ch = 2πrh
Where C is the circumference of the base, h is the height of the cylinder, and r is the radius of the base.
Total area of a cylinder = L.A. + 2𝐴𝑏𝑎𝑠𝑒
Cone: resembles a pyramid but with a circle base.
Theorem 114: The lateral area of a cone is equal to one-half the product of the slant height and the circumference of the
base.
𝐿𝐴𝑐𝑜𝑛𝑒 = ½ Cl = πrl
Where C is the circumference of the base, l is the slant height, and r is the radius of the base.
Spheres:
T.A. of a sphere = 4π𝑟 2
12.4 Volumes of Prisms and Cylinders
Postulate: The volume of a right rectangular prism is equal to the product of its length, its width, and its height.
Volume of a rectangular box = l*w*h
Where L is the length, w is the width, and h is the height.
Theorem 113: The volume of a right rectangular prism is equal to the product of the height and the area of the base.
Volume = height*B
Where B is the area of the base and h is the height.
Theorem 116: The volume of any prism is equal to the product of the height and the area of the base.
Volume = Bh
Theorem 117: The volume of a cylinder is equal to the product of the height and the area of the base.
Volume = Bh = π𝑟 2 h
Cross section: the intersection of a solid with a plane.
Theorem 118: The volume of a prism or a cylinder is equal to the product of the figure’s cross-sectional area and its
height. Volume = area of Cross-section*height
12.5 Volumes of Pyramids and Cones
1
Theorem 119: Volume of a pyramid = 3Bh
1
1
Theorem 120: Volume of a cone = 3Bh = 3 π𝑟 2 h
Theorem 121: In a pyramid or cone, the ratio of the area of a cross section to the area of the base equals the square of the
ratio of the figures’ respective distances from the vertex.
𝐴𝑟𝑒𝑎 𝑜𝑓 𝐶𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑣𝑒𝑟𝑡𝑒𝑥 2
=(
)
𝐴𝑟𝑒𝑎 𝑜𝑓 𝐵𝑎𝑠𝑒
ℎ
12.6 Volumes of Spheres
4
Theorem 122: The volume of a sphere = 3 π𝑟 3
PRACTICE PROBLEMS
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
10.)
11.)
SOLUTIONS
1.)
2.)
3.)
4.)
For x: 380/500 = 425/x  380x = 500*425
x= 212,500 / 380 = 559.2
For y: 380/600 = 425/y  380y = 600*425
y= 255,000 / 380 = 671.1
600 + 500 + 380 + 750 + 425 + 671.1 + 559.2 = 3,885.3 feet
5.)
6.)
7.)
8.)
Angle w = 180 – (110 +48) = 22
Angle x = 90 – 22 = 68
Angle y = 180 – (68+68) = 44
Angle z = 180 – y = arc UR.
180 – 44 = 136.
Angle z = ½*arc UR.
136/2 = 68.
9.)
10.)
11.)