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Math Magazine
Special Double Issue Edition
Area of Polygons and Circles
Circles
The Royal Wedding
Careers Using Circles and Area
Table of Contents
Page
Contents
3 to 15
16 to 24
Circles and Careers
Area of Polygons and Circles
and Careers
The Royal Wedding, Real-Life
Applications and Quiz with
Answer Key
25 to 30
Circles
In this section about circles you will learn about circles and their
properties. You will also learn about arcs on circles, angles and
segments related to circles, and how to graph circles.
Tangents to Circles
A circle is a set of points that are all equidistant from the center
of the circle. The distance from any of these points to the
center is the radius, circles are congruent if they have the
same radius. The distance across a circle, through the center,
is the diameter. A segment whose endpoints are on the circle
is a chord while a segment that intersects a circle in two points
is secant. A line that intersects the circle at one point is a
tangent. The point that the line intersects the circle is the point
of tangency. Circles that lie on the same plane and intersect at
one point are tangent circles while those who share a center
are concentric. If a line is tangent to coplanar circles it is called
a common tangent. Chiropractors who treat patients with
many problems including spinal injuries use circles thier work
and use their scientific knowledge.
Theorems on Tangents to Circles
Theorem 10.1 states that if a line is tangent to a circle, then it is
prependicular to the radius drawn to the point of tangency.
Theorem 10.2 states the in a plane, if a line is perpendicular to a
radius of a cirlce at its endpoint on the circle, then teh line is
tangent to the circle.
Theorem 10.3 states that if two segments from teh same
exterior point are tangent to a cirlce, then they are congruent.
Arcs and Chords
An angle whose vertex is the center of a circle is a central
angle. If the central angles measures less than 180 it creates a
minor arc on the circle, if it measures more than 180 it creates a
major arc. If the endpoints of an angle are on the diameter then
the arc is a semicircle. The measures of the arcs are equal to
the measures of the central angles that create them. Two arcs
can be congruent if they have the same measure and are the
same circle or congruent circles. There is one postulate
dealing with arcs, the Arc Addition Postulate. This states that
the measure of an arc formed by two adjacent acrs is the sum
of the measures of the two arcs. Astronomers use cirlces in
their mapping of the universes along with creativity.
Theorems on Arcs and Chords
Theorem 10.4 states the in the same circle, or in congruent
circles, two minor arcs are congruent if and only if their
corresponding chords are congruent.
Theorem 10.5 states that if a diameter of a cirlce is
perpendicualr to a chord, then the diameter bisects the chord
and its arc.
Theorem 10.6 states that if one chord is a perpendicular
bisector of another chord, then the first chord is a diameter.
Theorem 10.7 states that in the same circle, or in congruent
circles, two chords are congruent if and only if they are
equidistant from the center.
Inscribed Angles
Inscribed angles are angles whose vertex is on the circle and
whose sides contain chords of the circle. The arc that is
created by the angle is callled the intercepted arc. Polygons
whose vertices all lie on a cirlce are inscribed in the circle while
the circle is circumscribed about the polygon. Physicists study
the matter that makes up the universe and natural forces such
as gravity, they use cirlces in their studies on a regular
basis. Physicists also use scince and technology on regular
basis.
Theorems on Inscribed Angles
Theorem 10.8 states that if an angle is inscribed in a circle,
then its measure is half the measure of its intercepted arc.
Theorem 10.9 states that if two inscribed angles of a circle
intercept the same arc, then the angles are congruent.
Theorem 10.10 states that if a right triangles is inscribed in a
circle, then the hypotenuse is a diameter of the
circle. Conversely, if one side of an inscribed triangles is a
diameter of the circle, then the trianlges is a right triangle and
the angles opposite the diameter is the right angle.
Theorem 10.11 states that a quadrilateral can inscribed in a
circle if and only if its opposite angles are supplementary.
Other Angle Relationships in Circles
Two lines can intersect a circle in three different places. They
can intersect on the circle, inside the circle, or outside the
circle. Boilermakers use circles in making boilers, vats, and
other large tanks and innovation to create new methods of
building vats, ect.
Theorems on Other Relationships
Theorem 10.12 states that if a tangent and a chord intersect at
a point on a circle, then the measure of each angle formed is
one half the measure of its intercepted arc.
Theorem 10.13 states that if two intersect in the interior of a
circle, then the measure of each angle is one half the sum of
the measures of the arcs intercepted by the angle and its
vertical angle.
Theorem 10.14 states that if a tangent and a secant, two
tangents, or two secants intersect in the exterior of a cirlce,
then the measure of the angle formed is one half the difference
of the measures of the intercepted arcs.
Segment Lengths in Circles
Tangent segments are tangents that tangent to a cirlce at
their endpoint. Secant segments are the segments of a
secant that go from a given point to the opposite side of the
circle. The external secant segment is the part that is on the
exterior of the circle. Drafters use circles in their technical
drawings of spacecraft to office buildings they also use preenginneering teachings and creativity in their work.
Theorems on Segment Lengths
Theorem 10.15 states that if two chords intersect in the interior
of a circle, then the product of the lengths of the segments of
one chord is equal to the product of the lengths of the
segments of the other chord.
Theorem 10.16 states that if two secant segments share the
same endpoint outside a circle, then the product of the length of
one secant segment and the length of its external segment
equals the porduct of the length of its external segment.
Theorem 10.17 states that if a secant segment and a tangent
segment share an endpoint outside a circle, then the product of
the length of the secant segment and the length of its external
segment equals the square of the length of the tangent
segment.
Equations of Circles
When writing the equation of a circle the radius will be
represented by r, the center by (h,k), and any given point on the
circle by (x,y). You can then put this into the distance formula
and to find the standard equation of a circle you can square
both sides. This gives you (x-h)2+(y-k)2=r2. If the center of the
circle is the orgin then the equation is x2+y2=r2. Machinists,
who make specialized parts of advanced techonology, use
circles in their product frequently with technology.
Locus
Locus in a plane is the points that satisfy the given condition or
conditions. Locus is derived form the Latin word for
location. The plural for locus is loci. Compter software
engineers use circles, and loci, very often when designing
equations for software with math and science in unison with
innovation.
Area of Polygons and Circles
In this section about area of polygons and circles you will learn
about angle measures and areas of polygons. You will also
learn how to compare perimeters and areas of similar
figures. You will lastly learn how to the find the circumference
and area of a circle along with other measures.
Angle Measures in Polygons, Theorems
To understand angles measures in polygons there are two
theorems and corollaries. Theorem 11.1 states that the sum of
the measures of the interior angles of a convex n-gon is (n2)x180. The corollary to this theorem states that if the measure
of each interior angle of a regular n-gon is (1/n)x(n-2)x180, or
(n-2)x180/n. Theorem 11.2 states that the sum of the measures
of the exterior angles of a convex polyogn, one at each vertex,
is 360. The corollary to this theorem states that the measure of
each exterior angle of a regular n-gon is (1/n)x360, or
360/n. Landscape architects use polygons and their angle
measures to create appealing outdoor areas for clients with
creativity.
Areas of Regular Polygons
The center of a polygon and radius are the same as those of
the polygon's circumscribed circle. The apothem of a polygon
is the distance from the center to any side of the polygon. This
is also the height of a triangle made with the radii to the
consecutive vertices. The central angle of a polygon is an
angle that is formed with the center and two consecutive
radii. In a regular polygon a central angle measurement can
be found by dividing 360 by the number of sides. Real estate
agents use area to find the area in their houses and buildings
while using the 21st century skill of math.
Areas of Regular Polygons Theorems
Theorem 11.3 states that the area of an equilateral triangle is
one fourth the square of the length of the side times the square
root of three.
Theorem 11.4 states that the area of a regular n-gon with side
length s is half the product of hte apothem a and the perimeter
P.
Perimeters and Areas of Similar Figures Theorems
To understand the realtionship between perimeter and area of
similar figures you must understand this theorem.
Theorem 11.5 states that if two polygons are similar with the
lengths of corresponding sides in the ratio of a:b, then the ratio
of their areas is a2:b2.
Agricultural workers use area and perimeter frequently in their
farming and planning along with science and innovation.
Circumference and Arc Length
The circumference of a circle is the distance around the
circle. The ratio of the circumfernce to the diameter is pi. Arc
length is a given portion of the circumference of a circle. The
lengths of a semicircle and a 90 degree arc are half the
circumference and one fourth of the circumference,
respectively. Police and detectives use cirlces in their mapping
of areas for many different reasons along with creativity to solve
cases.
Circumference and Arc Length Theorems
Theorem 11.6 states the circumference C of a circle is
C=(pi)d or C=2(pi)r, where d is the diameter of the circle and
r is the radius of the circle.
Arc Length Corollary states that in a circle, the ratio of the
length of a given arc to the circumference is equal to the ratio
of the measure of the arc to 360 degrees.
Areas of Circles and Sectors Theorems
Theorem 11.7 states that the area of a circle is pi times the
square of the radius.
A sector is a region of a circle bound by two radii and their
intercepted arc.
Theorem 11.8 states that the ratio of the area of a sector of a
circle to the area of the circle is equal to the area of the
measure of the intercepted arc to 360 degrees.
Aerospace engineers use circles (and math, science, and preengineering) in their work designing, contructing, and testing
different types of aircraft while using creativity, technology, and
innovation.
Geometric Probability
Probability is the likeness of an event to occur represented by a
number from zero to one. Geometric probability is a version of
probability that involves length and area. Atmospheric
scientists, meteorologists, use geometric probability in their
predicition of where events will occur along with technology.
Royal Wedding Quiz
This is the Duke and Duchess
of Cambridge's wedding
cake. If Catherine slices a
piece from the center of the top
layer with an anlge of 45
degrees what is the measure of
the arc that is the outside of
the piece of cake?
Supposed that the area of the
piece of cake is 14 and one
eighth of a square inch, what is
the area of the top layer?
Answers
45 degrees
113 square inches
From the center of the Duchess
of Cambridge's bouquet to the
outer edge there are only type
of flowers ina certain area. The
whole area of the bouquet is 7
inches and the arc length of the
given area is 80 degrees. What
is the area of this sector?
Answer
1.5 square inches
This is the a section of the
floor in Westminster Abbey
where the marriage ceremony
took place. What is the
geometric probability that the
ceiling will collapse above the
smaller square if it has side
lengths of 14 feet inside the
larger square with side
lengths of 20 feet?
Answer
49%
Identify where the central
angle, radius, and apothem
of this window in
Westminster Abbey.
Answer
These photographs show
the soon to be Duchess of
Cambridge riding in the
Queen's Rolls-Royce
Phantom VI and Prince
William through similar
quadrilateral windows. If
the ratio of sides is 2:3,
what is the ratio of areas of
the windows?
Answer
4:9
Created by Carly Cannoy