Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
Geometry POW Solution, November 15 (part 2)
Posted:
Dec 21, 1996 11:50 AM
|
|
***********************************************
From: LISHACK@SASD.K12.PA.US
From: Becky Spinella and Erin May
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
PART 1
WE FOUND THE RADIUS OF THE CIRCLE WHICH IS 6 TIMES THE
SQUARE ROOT OF TWO. THIS EQUALS 8.49. TO FIND THE CIRCUMFERENCE
OF A CIRCLE, USE THE EQUATION, 2(PI)R. SUBSTITUTE 8.49 FOR R.
THE ANSWER WOULD BE 53.31. WE DECIDED THAT A CIRCLE IS 360
DEGREES. IF WE DIVIDED 360 DEGREES BY 90 DEGREES, WHICH IS A
RIGHT ANGLE, WE WOULD FIND THAT A CIRCLE IS DIVIDED INTO FOUR
QUARTERS. DIVIDE THE CIRCUMFERENCE BY 4, AND GET THE ANSWER
AS 13.33.
PART 2
AN EQUILATERAL TRIANGLE HAS 3 EQUAL SIDES, AND THREE 60
DEGREE ANGLES. DIVIDE 360 DEGREES BY 60 DEGREES AS IN PART ONE.
THIS WILL EQUAL SIX TRIANGLES. DIVIDE THE CIRCUMFERENCE BY THE
SIX TRIANGLES AND GET THE ANSWER 8.885.
PART 3
FOR X DEGREES, SUBSTITUTE X IN FOR WHEREVER THE AMOUNT
OF DEGREES WOULD BE IN THE EQUATION. IN THIS SITUATION, DIVIDE
360 DEGREES BY X DEGREES. THEN, DIVIDE THE CIRCUMFERENCE BY
360/X DEGREES. THE ANSWER WOULD BE 53.31 DIVIDED BY 360/X
DEGREES. ANOTHER WAY OF SOING THIS WOULD BE MULTIPLYING 53.31
BY X DEGREES OVER 360 DEGREES. THIS EQUALS THE SUBTENDED ARC
OF ANGLE X.
***********************************************
From: Lishack@SASD.K12.PA.US
From: Adam Yacono and Gretchen Schneider
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
In order to solve this problem you must know the formula to find
the length of a minor arc.
m=the measure of the angle of an arc, r=radius, L=length
L=m/180*pi*r
For the isosceles right triangle m=90 and r=6*sqrt(2). Substitute
these into the formula.
L=90/180*pi[6*sqrt(2)]
Simplified into terms of pi, the answer is L=pi(3*sqrt[2])
For the equilateral triangle each angle is equal to 60 degrees
because each angle is equilangular.
m=60, r=6*sqrt(2)
Then you substitute these into the equation to find...
L=60/180*pi[6*sqrt(2)]
This simplifies into L=pi*2*sqrt(2).
If the measure of the angle is x then you substitute this into
the formula.
L=x/180*pi[6*sqrt(2)]
This simplifies into L=x/30*pi*sqrt(2).
***********************************************
From: 74620.2745@compuserve.com
From: Kristen Pirozzi and Jessica Thompson
Grade: 9
School: Roselle Park High School, Roselle Park, New Jersey
The last one was really from SYRUP DESAI - sorry! I got their papers mixed up.
C=2Pi*r=2(3.14)(8.49)=53.34
There will be 4 right triangles in the circle.
arc is 53.34/4= 13.33
There will be 6 equilateral triangles in the circle.
arc is 53.34/6=8.89
There will be 360/x triangles in the circle.
arc is 53.34/(360/x)
***********************************************
From: khaire@khsd.k12.ca.us
From: Kyle Haire and Trevor Glasgow
Grade: 10
School: South High School, Bakersfield, California
Since the radius is 6 times the square root of 2, we first put it
into the circumference equation which is 2 times pie times r. The
circumference turned out to be 53.31. Since the non base angle is 90 degrees
it is one fourth of the circumference so we went 1/4 times 53.31 because
the arc
length equation simplified is
x/360 times circumference so the arc length was 13.32
If it were an equilateral triangle the non base angle would be
60 degrees. So 60 degrees is one sixth of the circumference so we put 1/6 times
53.31. It turned out to be 8.885.
If the non base angle was x, we would just have to replace the 60 degrees
with x
so it would be x/360 times 53.31
***********************************************
From: forostoski@worldnet.att.net
From: Nicole Forostoski
Grade: 10
School: Martin County High School, Stuart, Florida
Problem of the week November 11-15
Nicole Forostoski
Martin County High School
Stuart, Florida
Mrs. Summers Per. 6
Dear Annie,
"What is the lenght of the minor arc subtended by the cord?"
- 13.32192
The length of the minor arc subtended by the cord is 1/4th of
the circumference of the circle, because a right triangle is 90
degrees and a circle is 360 degrees and 90 is 1/4th of 360.
"What if it were an equilateral triangle instead of just
isosceles?" - 8.881280667
The length of the minor arch subtended by the cord is 1/6th of
the circmference of the circle, because you can draw 6
equilateral triangles inside the circle.
"What if I told you that the non-base angle was x degrees? Then
what would the answer be?"
You would divide 360 degrees by x degrees then you would divide
53.287684 (the circumfrence of the circle) by the answer to get
the length of the minor arc subtended by the cord.
For example if x is 10 degrees:
360/10=36
3.287684/36=1.480213444
the length of the minor arc is
1.480213444
***********************************************
From: jimsoil@aol.com
From: Jessica
Grade: 10
School: South High School, Bakersfield, California
In response to the problem of the week for november 11-15:
The length of the minor arc subtended by the chord is 13.336. I arrived
at this
answer by using the following formula: C=D(3.14) plug in the diameter which is
16.98 and you get 53.34 then since 90 degrees is 1/4 of a circle mutltiply
53.34
by 1/4.
If the triangle were equilateral then the answer would change and be 8.89. I
arrived at this answer by the same formula: C=D(3.14) plug in the diameter of
16.98 and you get 53.35 and the angle has changed because the triangle has, 60
degrees is 1/6 of a circle so multiply 53.34 by 1/6.
If the non- base angle was x then the formulas would still be the same except
the answer would be: 53.34 * x/360 degrees and that is how you arrive at any
answer relating to this circle.
Thank you for the opportunity to work this problem out my father and I have
never had so much fun!
***********************************************
From: lishack@sasd.k12.pa.us
From: Jen Baer and Brett Bernardo
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
The answers:
- length of the minor arc subtemded by the chord - 13.32864 units
- if it were an equilateral triangle instead - 8.88 units
- if it was X then it would be (X/(180)) * 3.14159 * sqrt(72)
The formula that the measurements are placed into is the formula in part
three of the solutions. Here, X is the measurement of the angle in the center
of the circle whose vertex is C. The sqrt(72) is the radius of the circle. To
find the first answer, substitute 90 in for X, and for the second solution
substitute 60 for X while leaving the rest of the equation alone. This
produces
the answers.
***********************************************
From: lishack@sasd.k12.pa.us
From: Gretchen Ross and Pranav Shetty
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
1) First you find the circumfrence of a circle by using the
formula 2(3.14)r. Put the numbers from the question in and you get 53.3
units.
Then since it is a isosceles right triangle it makes a 90 degree angle so it
cuts off one-fourth of the circle so you multiply the circumfrence 53.3*1/4=
13.33-lenght of minor arc.
2) You would use the same method but measure the angle in the equalateral
triangle which is 60 degrees and you would multiply 53.3 the circumfrence by 60
degrees/360 degrees or 1/6=8.88- lenght of minor arc.
3) You would use the same method but multiply the circumfrence by x/360 degrees
which would be the fraction of the circle which would be cut off by the
triangle.
***********************************************
From: Lishack@sasd.k12.pa.us
From: Melissa Antoszewski and Lynn DeLuca
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
Some of the terms used in this problem were very difficult to understand,
so we took the librety of looking them up so that we could solve the problem.
We found that a chord was a segment whose end points lie on a shpere and that
the hypotenuse is the side of a right triangle oppositie the right angle. We
also know that an isosceles triangle is one that has two congruent sides. In
other words it is the longest leg. We drew a diagram of the problem and made
the radii the lenghts of the legs. So, we used the equation: L = q/180 * pi *
radius. The radius given was applied to the equation along with the degree of
the triangle measure, which gave us the lenght of the subtendon arc.
We continued using this equation for the other questions, receiving
approximately13.5 units, and 8.89 units.
***********************************************
From: lishack@sasd.k12.pa.us
From: Greg Moore and Shawn Phelen
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
The base angle in the measurment of revolutions times the circumference of the
circle is equal to the minor arc subtended by the chord formed by the triangle
with the base angle as one of its vertii. Using this we can form an equation:
x/360 = revolutions of base angle where
"x" is equal to degrees of angle.
2 * r * Pi = Circumference of the circle
x x*r*Pi the length of the minor arc
--- * 2*r*Pi = -------- = where "x" is equal to the
360 180 base angle in degrees
In the problem "r" is equal to 6 * 2^.5. In the problem "x" is equal to 90
degrees, 60 degrees and x degrees.
answer 1 = 3 * 2^.5 * Pi = 13.328649
answer 2 = 2 * 2^.5 * Pi = 8.8857659
answer 3 = 2^.5*Pi*x
---------
30
***********************************************
From: Lishack@SASD.k12.PA.US
From: Andrew Miller and Sarah Kost
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
We first sketched the figure described. We gave the legs of
the triangle 6 times the square root of 2. We labeled the minor
arc 1/4x. Because there are 4 minor arcs in the entire circle,
we found the circumference of the circle or x. We used the
equation 2(pi)r. R or the radius = 6 times the square root of 2.
The circumference(x) is 53.3. 1/4(53.3) equals 13.3. Therefore
the length of the minor arc(1/4x) would 13.3 units.
If the triangle was an equalateral triangle with sides of 6
times the square root of 2, then the circle would be composed of
6 equalateral triangles, with 6 minor arcs. Find the
cicumference of the circle, which would be the same as before,
53.3. This time the minor arc would be 1/6x and the circle would
again be x. Divide 53.3 by 6 and the length of the minor arc
would be 8.89 units.
If the non-base angle was x, the equation would be 2(pi)r *
(x/360) which would equal 53.3x/360.
***********************************************
From: LISHACK@SASD.K12.PA.US
From: Katie Behling and Chris Vendilli
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
FIRST WE DEFINED THE UNKNOWN TERMS:
ISOCELES = TWO CONGRUENT SIDES + BASE, THE ANGLE OPPOSITE THE
BASE IS THE VERTEX
CHORD OF A CIRCLE = SEGMENT WHOSE ENDPOINTS LIE ON THE CIRCLE
HYPOTENUSE = THE SIDE OPPOSITE OF THE RIGHT ANGLE
RADII = PLURAL OF RADIUS
RADIUS = HALF THE DIAMETER OF THE CIRCLE
MINOR ARC = THE SMALLER ARC DIVIDED BY THE CHORD
SUBTEND = THE PORTION OF THE ARC DIVIDED BY THE CHORD
THE FIRST PART OF THE PROBLEM WAS TO FIND THE LENGTH OF THE MINOR
ARC SUBTENDED BY THE CHORD.
WE KNOW THAT THE RADIUS OF THE CIRCLE IS 6 SQUARE ROOTS OF 2,
WHICH IS APPROXIMATELY 8.5. USING THE FORMULA:
L OF ARC = q/180 * PI * r
LENGTH OF THE ARC = 90/180 * PI * 8.5 = 13.35
OUR SOLUTION IS 13.35
IN AN EQUILATERAL TRIANGLE ALL THE ANGLES ARE 60 DEGREES
PUTTING THIS IN THE EQUATION
L OF ARC = 60/180 * PI * 8.5 = 8.9
OUR SOLUTION FOR THE EQUILATERAL TRIANGLE QUESTION IS 8.9
THE NEXT QUESTION WAS TO FIND THE LENGTH OF THE ARC IF THE NON-
BASE ANGLE WAS X, USING THE EQUATION:
L OF ARC = X/180 * PI * 8.5 = 26.7X/180
26.7X/180 = .15X
OUR SOLUTION FOR THE NON BASE ANGLE IS: .15X
I THINK AND HOPE.
I ANSWERED ALL OF THE PARTS.
***********************************************
From: LISHACK@SASD.k12.PA.US
From: Lisa Jasneski and Richard Erb
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
YOU MUST KNOW THAT A TRIANGLE HAS 180 DEGREES. THE RADIUS IS 6 TIMES THE
SQUARE
ROOT OF 2 OR 8.4852. AS YOU ALL KNOW, A CIRCLE CONTAINS 360 DEGREES.
THEREFORE, I DIVIDED 360 BY 90(THE MEASURE OF THE VERTEX ANGLE) AND GOT AN
ANSWER OF 4. THAT MEANS THAT THE LENGTH OF THE ARC FROM THE TWO POINTS ON THE
CIRCLE IS 1/4 THE CIRCUMFERENCE OF THE CIRCLE. THE FORMULA OF THE
CIRCUMFERENCE
OF A CIRCLE IS PI*D. GOING ON THAT, I MULTIPLIED 3.1416 BY 8.4852*2(BECAUSE
DIAMETER IS RADIUS TIMES 2) AND GOT 53.3142. BEING THAT THE LENGTH OF THE ARC
IS 1/4 THE CIRCUMFERENCE, I DIVIDED MY CIRCUMFERENCE(8.4852) BY 4. MY
ANSWER IS
13.3286 UNITS.
I USED THE SAME IDEA FOR THIS ONE TOO. I DIVIDED 360 BY 60(BECAUSE THERE
ARE 3
60 DEGREE ANGLES IN AN EQUILATERAL TRIANGLE) AND GOT 6. THEREFORE, THE LENGTH
OF THE ARC IS 1/6 THE CIRCUMFERENCE. USING MY CIRCUMFERENCE OF 53.3142
UNITS, I
DIVIDED 6 INTO IT. I GOT AN ANSWER OF 8.8857 UNITS.
STILL USING THE IDEA, I DIVIDED 360 BY X. THAT GOT A RESPONSE OF 360/X.
I THEN
DIVIDED 53.3142 BY 360/X OR 53.3142 * X/360. THAT GAVE A FINAL ANSWER OF
.1481X
FOR THE LENGTH OF THE ARC.
***********************************************
From: Lishack@sasd.k12.pa.us
From: Keith Dougall and Jenny Booth
Grade: 10
School: Shaler Area High School, Pittsburgh, Pennsylvania
The length of the minor arch in the first part will be 13.33.
We found this by first finding the distance of the radians to be
8.49. This was given to us in the problem in equation form, but
we converted it to decimal form. We then drew a diagram to
illustrate the problem. We found that four triangles would fit
in the circle. This means we would find the circumference and
then multiply it by 1/4. We got the circumference of 53.3.
Then we multiplied it by 1/4 and got the length of the minor arch
to be 13.33.
For the second part we did the same thing but multiplied the
circumference by 1/6 because six triangles would fit in the
circle. The answer we found for the minor arch was 8.88.
For the third part, we multiplied the circumference by x/360
because that is another way to find how many triangles would fit
in the circle. The length of the minor arch would now be
53.3x/360.
***********************************************
From: Kurt_Davies@shhs1.ccsd.k12.co.us
From: Kurt Davies
Grade: 10
School: Smoky Hill High School, Aurora, Colorado
The length of an arc is equal to the circumference of its cicle
times the fractional part of the circle determined by the arc.
So the first thing I did was find the circumference of the circle.
This was twice the radius (6 times the square root of 2) times pie.
This equaled 53.314. Then I need to multiply this by 90/360 for
the length of the first arc. 53.314 times 90/360= 13.32. The next
triangle was an eauilateral triangle so the angle had to be 60.
To find the length of the arc I multiplied 53.314 times 60/360.
This equaled 8.885. Since the third angle was x the length of the
arc would be 53.314 times x/360. This equlaed 53.314x/360
***********************************************
From: Mike_Rogers@shhs1.ccsd.k12.co.us
From: Mike Rogers
Grade: 9
School: Smoky Hill High School, Aurora, Colorado
If the angle of the vertex is 90 degrees then the distance of the minor arch
would be 13.32864881
If the angle of the vertex is 60 (equilateral triangle) degrees then the
distance of the minor arch would be 8.885765876
If the angle of the vertex is x degrees then the distance of the minor arch
would be 53.31459526/(360/x)
***********************************************
From: mshulman@lausd.k12.ca.us
From: Michael Shulman
Grade: 11
School: North Hollywood High School, Hollywood, California
First, the length of an arc of a circle is equal to the
circumference of the circle multiplied by the fraction of the
circle that the arc is. This fraction is equal to the number
of degrees in the central angle of the arc divided by 360.
Therefore, since the circumference (C=2*pi*r) of this circle is
12*pi*sqroot(2), the answers are:
Part I: C*(90/360)=C/4= 3*pi*sqroot(2)
Part II: C*(60/360)=C/6= 2*pi*sqroot(2)
Part III: C*(x/360)= x*pi*sqroot(2)/30
***********************************************
From: mojave@ridgecrest.ca.us
From: Cassie Gorish
Grade: 9
School: Burroughs High School, Ridgecrest, California
Answer #1:
Since 90 degrees is one quarter of a circle, the length of the arc
is one quarter of the circumference.
C = pi times d
C = pi times 2 times six squareroots of 2
C = approximately 53.31
C divided by 4 = approximately 13.33, which is the length of the
arc in this case.
Answer #2:
Since _60_ degrees is one sixth of the circle, the length of the
arc is one sixth of the circumference.
C = pi times d
C = pi times 2 times six squareroots of 2
C = approximately 53.31
C divided by 6 = approximately 8.89, which is the length of the
arc in this case.
Answer #3:
Since x degrees is a {x over 360} part of the circle, the
length of the arc in this case is that part of the circumference.
______C_______
360 divided by x
***********************************************
From: mongsys@iponline.com
From: Roger Mong
Grade: 8
School: Zion Heights Junior High School, Richmond Hill, Ontario, Canada
1. The legs are the radii means that the right angle is at the
centre of the circle. Since A right angle is a quater of a
full circle, the minor arc would then be a quarter of the
circumference.
6 * 2^* * 2 * 3.14 * *
= 8.49 * 2 * 3.14 * .25
= 13.33
2. Same thing as #1 but there would be a 60ð angle in the center
of the circle instead, because an equilateral triangle has all
of its angles equals to 60 degrees. So the minor arc would be
a sixth of the circumference since 60 = 360/6
6 * 2^* * 2 * 3.14 * 1/6
= 8.49 * 2 * 3.14 * 0.17
= 8.89
3.Same thing as before: if the angle at the centre is x degrees,
then the minor arc would be the circumference multipy thee
percentage that x degrees takes up in a full circle.
6 * 2^* * 2 * 3.14 * x/360
= 8.49 * 2 * 3.14 * x/360
= 53.31 * x/360
= 0.1481 * x
***********************************************
From: Mona165278@AOL
From: Sheridan Waller
Grade: 10
School: Smoky Hill High School, Aurora, Colorado
The measure of a right angle is 90 degrees.By definition, the
measure of a minor arc is the same as the measure of the central
angle that intercepts the arc.In other words, the measure of the
central angle is the same as the minor arc. So if the central
angle is 90 degrees, so is the minor arc. Since there are 360
degrees in a circle, 90/360 is 1/4. The minor arc is 1/4 of the
circle. The length,however, would be 1/4 times the circumfrence.
The circumfrence equation is 2(pi)r.If the radius is 6 times the
square root of 2, the circumfrence is 2(pi)6 times the square root
of 2,which can be simplified into 12 times the sqaure root of
2(pi). When multiplied by 1/4, the 4 cancels out and the answer
is 3 times the square root of 2(pi). If the triangle was
equilateral, all angles would be 60 degrees.60/360 reduces to 1/6.
When multiplied by the circumfrence, the answer is 2 times the square
root of 2(pi). If x/360 is multiplied by the circumfrence, the answer is x
times
the square root of 2(pi)/30.
***********************************************
From: ald@pacbell.net
From: Jennifer Kaplan
Grade: 6
School: Castilleja Middle School, San Francisco, California
1. I found that the right triangle had to be 1/4 of the circle,
therefore the length of the minor subtending arch would be
1/4 of the circumference of the circle
(12*square-root-of-2*pi), so the length of the minor subtending
arch is 3*square-root-of-2*pi!!!!!
2. If the triangle is equilateral than the length of the
minor subtending arch is 1/6 of the circumference of
the circle (thinking back to of the previous POW
problems the hexagon in side a circle . . ), therefore
if the triangle is equilateral than the length of the subtending
arch is 2*the-square-root-of-2*pi!!!!!!!
3. If you know that the non-base angle is x you can
set up a proportion, x is to the minor subtending arch length(y)
as 360 is to 12*square-root-of-2*pi. Therefore 360y =
12*square-root-of-2*pi*x so,
12*square-root-of-2*pi*x/360 = y, therefore the length of the
minor subtending arch is square-root-of-2*pi*x/30!!!!!!!
***********************************************
From: rljnjld@sierratel.com
From: Bob Jackson
Grade:
School:
Subject: Re: [Fwd: Geometry Problem of the Week, November 11-15]
> A chord of a circle is the hypotenuse of an isosceles right triangle whose
> legs are radii of the circle. The radius of the circle is 6 times the
> square root of 2. What is the length of the minor arc subtended by the
> chord?
chord = C/4
C=2*pi*D
C = 12*pi*sqrt(2)
chord = 3*pi*sqrt(2)
>
> What if it were an equilateral triangle instead of just isosceles?
chord = C/6
C=2*pi*D
C = 12*pi*sqrt(2)
chord = 2*pi*sqrt(2)
>
> What if I told you that the non-base angle (so the angle at the vertex at
> the center of the circle) was x degrees? Then what would the answer be?
chord = C/(360/angle)
C=2*pi*D
C = 12*pi*sqrt(2)
chord = (12*pi*sqrt(2))/(360/angle)
--
-------------------------------------------------------------------
Bob Jackson beeej@juno.com rljnjld@sierratel.com
10th grade dropout
-------------------------------------------------------------------
***********************************************
From: roderick@teleport.com
From: Gavin Calkins
Grade: 8
School: Waluga Junior High School, Portland, Oregon
Subject: November 11-15 Problem of the week
Waluga Jr. High
Grade 8
11-12-96
I used Geometer's Sketchpad for the graphics: maybe it will work this
time.
~Gavin Calkins
This right isosceles triangle formed a 90 degree angle at the center of the
circle with its sides intersecting the circle to form the arc that we need to
measure. Since there are 360 degrees in a circle, the minor arc is 90/360 or 1/
4 of the circumference. Since the radius of the circle = 6 (radical)
2, the circumference can be found using the formula 2( pi) r. Therefore,
2(pi)(6 radical 2) = 12(pi)(radical 2) or approximately 53.315. Then the
arc is
1/4 of 53.315, which is 13.33 or 3(pi)(radical 2) exactly.
In an equilateral triangle, each angle measures 60 degrees. Therefore, the
center angle that determines the arc also measures 60 degrees. This is 60/360
or 1/6 of the circumference of the circle. The radius remains 6( radical 2).
Using the formula for the circumference, 2( pi) r, the circumference of the
circle would be 2 (pi)( 6)( radical 2) which equals 12 (pi)(radical 2) or
approximately 53.315. Since the arc is 1/6 of the circumference, its measure =
(53.315)/6 = 8.8858 or 2 (pi)(radical 2) exactly.
If you look back at the first two problems, you can find a pattern for finding
the measure of the arc formed by the center angle. For the right angle, the
formula for the measure of the arc formed
was
(90/360)(12)pi(radical 2)
For the equilateral triangle, the arc formed by the 60 degree center angle
was
(60/360)(12)pi(radical 2)
You can conclude from this that any center angle of this circle with a measure
X will form an arc that measures
(X/360)(12)pi(radical 2)
Therefore, to find the arc measure formed by any center angle of a circle,
first
form a fraction of the center angle measure over 360 degrees (total degree
measure of a circle). Then multiply that fraction times the measure of the
circumference of the circle found by using the formula 2(pi)r.
***********************************************
From: kheldt@tcoe.trinity.k12.ca.us
From: Terry Haslam
Grade: 11
School: Southern Trinity High School, Mad River, California
Subject: pow
Terry got
3 times the square root of 2 times pi
2 times the square root of 2 times pi
The square root of 2 times pi times x all over 30
He used simple geometry concepts and the formula Circumference=2rpi to
solve them.
***********************************************
From: ruth@forum.swarthmore.edu
From: Becky Dunlap and Lauren Kupersmith and Swathi Bala
Grade: 9
School: Germantown Academy, Fort Washington, Pennsylvania
Subject: GA pow
Becky Dunlap, Lauren Kupersmith, and Swathi Bala, 9th Grade
Mrs. Carver's Geometry Honors Class
Germantown Academy
Fort Washington, PA
The minor arc is 90 degrees because the degrees of the minor arc are deterined
by the degrees of the angle CAB inside the circle displayed above. Since 90
degrees is one quarter of the circle (90 degrees divided by 360 degrees) the
equation for finding the length of the minor arc is as follows: (1/4) x (the
circumference). The circumference is 2(pi)r. In this case, r is 6 root 2 so
the circumference of the circle is 53.3. The length of arc CB is (1/4)(53.3)
which = 13.3.
An equilateral triangle can't be a right triangle, all angles are 60 degrees.
So the degrees of the minor arc are 60 degrees. 60 degrees is 1/6 of the
circle. Your equation for the mionor arc now is (1/6)(53.3) because the
circumference is the same. The length of arc CB is 8.9.
If the non base angle was "x" degrees and the two sides were still 6 root 2,
your minor arc would be (x/360)(53.3) which is 53.3x/360. But if the sides are
not 6 root 2 and you didn't know what the radius was, the equation would be (x/
360)(2(pi)r) so that is the same as (pi)r(x)/180.
These numbers are all rounded to the nearest tenth. Thank you have a nice week!!
***********************************************
From: EGRIFF00@brookstone.ga.net
From: Adam Hobson
Grade:
School: Brookstone School, Columbus, Georgia
Subject: problem of the week November 11-15
Problem of the Week, November 11-15
By multiplying 6 times the square root of 2, the radius, by 2, the
diameter of the circle is found. The diameter, which comes out to be
16.971, is then multiplied by Pi to get the circumference, 53.315.
Because the right triangle takes up 90 degrees of the 360 degrees
circle, or one fourth of it, the length of the minor arc is one
fourth the circumference. When the circumference is divided by four
13.329 is found as the length of the minor arc.
If the triangle is equilateral, all the angles are also equal. When
180, the total degrees in a triangle, is divided by three, the number
of angles in a triangle, we find that each angle is 60 degrees. When
60 is divided by 360, we see that the triangle takes up 1/6 of the
circle. When 53.315, the circumference, is multiplied by 1/6, the
fraction of the circle the triangle takes up, we find that the minor
arc is 8.886 units long
To find the length, you first take the circumference of the circle:
53.315. You then divide x by 360 (the total number of degrees in a
circle) to find what fraction of the circle the minor arc takes up.
53.315 is then multiplied by x/360. The formula to find the length
of the arc is 53.315*(x/360). When simplified, this is 53.315x/360.
Adam Hobson
Brookstone School
Columbus, GA
Geometry, Chappelle
***********************************************
From: anonymous@anonymous.com
From: YA
Grade:
School: Smoky Hill High School, Aurora, Colorado
Subject:
The Measure of an arc is equivalent to the number of degrees it occupies. A
complete circle occupies 360 degrees.
1) Since the measure of the arc is 90, it's lenght (l) can be expressed as
one fourth of the circle's circumference.
l=90C/360 C=2nr
l=(1/4)2nr=13.32
2)If it were an equilateral triangle instead of just isosceles the measure of
the arc would be 60, because an equilateral triangle has each angle equals to
60 degrees.So 60C/360=8.88
3)If the non- base angle was x degrees, then l=xC/360=(x)0.15
***********************************************
From: NormOrn@aol.com
From: Daniel Ornstein
Grade: 8
School: Georgetown Day School, Washington, DC
Daniel Ornstein
November 14, 1996
Washington D.C.
Georgetown Day School
Paul Nass
After a while of experimenting around with this experiment, I
figured out a
formula which worked for each of the three scenarios.
First of all, we know the formula for finding the circumfrance:
c=2(r
So, I figured since a triangle has 360 degrees, the amount of degrees in the
angle of the triangle which is at the vertex of the circle determines what
percentage of the circumfrance the arc is. Let ëXàreprisent the angle:
2(r
360(X
So now, we have our formula, and all that we need to do now is to test it and
make sure that it works, and then we will have the answers to the questions:
for the first problem:
we know that the radius is 6(2,
and that the vertex angle is 90(
so, our equation is:
2(6(2
360(90
12((2
4
3((2
And we arrive at the answer to the first question. If we follow the same
procedure for the next problem, we can find the answer:
We know the radius is 6(2, and the vertex angle is 60( so...
2(6(2
360(60
12((2
6
2((2
And lastly, we come to the most difficult question, and we still use the same
formula...
We still know the radius is 6(2, and the vertex angle this time is ëXÃÂ...
2(6(2
360(X
12((2
360(X
And even though this isnÃÂt that much of a specific answer, it is the best that
we can do.
And that is how I figured out this weekÃÂs problem of the week!
***********************************************
From: star1110@gnn.com
From: Hector Mercedes
Grade: 12
School: George Wingate HS, Brooklyn, New York
Subject: POW Nov 11 - 15
Hector
Mercedes
George Wingate H.S.
Grade
12
Brooklyn, NY
I set up a diagram the way it was explained to me in this first question. This
is how the diagram looks like:
Then I found the circumference
of this circle
C = 2pi r
C = 2(3.14...)(6\/2)
C = (6.28)(6\/2)
C= 53.31
Afterwards I set up a ratio proportion
360 : 90 = 53.13 : x
360x = 4797.9
x = 13.3275
The answer estimated to the hundredth of a unit 13.33
The second one, I did the same thing, except I placed an isosceles trianglein
place of it.
360 : 60 = 53.31 : x
360x = 3198.6
x = 8.885
For the last one, I had to use a bit of
inginuity ( 2 pi r) / 360/x
(see Hector's attachment, pow11.gsp)
|
|
|
|
