# Questions on Geometry: Polygons answered by real tutors!

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Question 994114: Number of sides of two polygons differ by 3. If their exterior angles differ by 20 then find the number of sides of each polygon

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Solve for , then calculate

John

My calculator said it, I believe it, that settles it

Question 993325: Prove that the angles between adjacent diagonals at any vertex of an n-sided regular polygon are equal and have the value 180/n.
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.
It is so because these angles are leaning on equal arcs, and each such arc has the arc measure of 360°/n.

Question 992982: Each figure is a regular polygon. Expressions are given for two side lengths. Find the value of x.
a square: one side x2 + x and another side: x2 + 4
a hexagon: one side: x2 + 3x and another side x2 + x + 2

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a square have same sides:

sides are:

a hexagon: all sides are equal

check:

Question 992749: The ratio of the interior angle to the exterior angle of a regular polygon is 5:2. Find the number of the sides of the polygon
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The ratio of the interior angle to the exterior angle of a regular polygon is 5:2. Find the number of the sides of the polygon
================
5x + 2x = 180
x = 180/7
----
Ext angles = 360/7
n = 360/(360/7) = 7 sides

Question 992503: ST is one side of the polygon and O is the center. What is the size of < SOT?
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Measure of angle SOT = 360/n

where n = number of sides

Note: this is assuming you have a regular polygon.

Question 990236: Seven of the interior angles of a nonagon add up to 1020° and one of the remaining angles is twice the other. Find the size of each remaining angle

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one of the angles will be 80 degrees.
the other angle will be 160 degrees.
the sum of the angles of the nonagon will be equal to 1020 + 240 = 1260.

a nonagon has 9 sides.
the sum of the angles of the nonagon are equal to (9-2)*180 = 7 * 180 = 1260.
the sum of the angles of a nonagon is also equal to 9 * (180 - 360 / 9) which is equal to 9 * 140 which is equal to 1260.
both formulas lead to the same sum of angle of a nonagon (nine sided figure).
once you know that, it's a simple formula to derive the angles.
let one of the angle = x and the other angle = 2x.
their sum is 3x.
1020 + 3x = 1260
3x = 1260 - 1020
3x = 240
x = 80.
2x = 160.

Question 990044: the sum of the outer angles of a polygon is twice the sum of the inner angles?how many sides does it have?what if the sum of outer angles is half the sum of inner angles? and if the sums are equal?
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the sum of the outer angles of a polygon is twice the sum of the inner angles?how many sides does it have?
---
The sum of the exterior angles is always 360 degs
--> interior angles = 180 --> a triangle.
-----------------
what if the sum of outer angles is half the sum of inner angles?
Do the same way as above.
--------------
and if the sums are equal
See above.

Question 988767: find the perimeter of the regular polygon
given information:
one side is 5(x+2) and the other is 7x+4
Thanks

Found 2 solutions by MathTherapy, josgarithmetic:
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find the perimeter of the regular polygon
given information:
one side is 5(x+2) and the other is 7x+4
Thanks
No polygon - regular or irregular - has 2 sides. Polygons start with 3.



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You NEED to know what kind of regular polygon, specifically HOW MANY SIDES, otherwise you can NOT find the perimeter.

The polygon being "regular", means the sides are congruent (same or equal measures), so at least you can say , and if the solution to this is reasonable, then you can find just the size of a side.

You only know from this that each side length is
------how many units is one side.

Question 987678: Let x represent the side length of a square. Find a regular polygon with side length x whose perimeter is twice the perimeter of the square. Find a regular polygon with side length x whose perimeter is three times the length of the square.
I just can't figure out how to calculate that when I don't know what type of polygon and how many sides it has?? Thanks for any help

Found 2 solutions by macston, solver91311:
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x=length of side; n=number of sides
perimeter=(number of sides)(length of side)
Perimeter=nx
perimeter(square)=4x
Regular polygon with perimeter 2 times square:
perimeter=2 times perimeter of square=2(4x)=8x
Perimeter=nx
8x=nx
8=n Number of sides=8
ANSWER 1: The polygon is an octagon.
.
length of square=x
perimeter=3x
perimeter=nx
3x=nx
n=3 Number of sides=3
ANSWER 2: The polygon is a triangle.

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If the side lengths are the same, the only way for a polygon to have a perimeter twice as large as the square is for the polygon to have twice as many sides as the square. There are 4 sides on a square, so the polygon with twice the perimeter must have how many sides?

John

My calculator said it, I believe it, that settles it

Question 985738: if the exterior angle of a regular polygon is 55degrees what is the value of its interior angles
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if the exterior angle of a regular polygon is 55degrees what is the value of its interior angles
-------------
1st, that's not possible.
Int = 180 - Ext = 125
-------------------
360/55 is not an integer --> no such regular polygon is possible.

Question 985588: What is the angle of a 500 sided polygon
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What is the angle of a 500 sided polygon
----------------
If it's a regular polygon, each exterior is 360/500 degs.
The interiors are 180 - Ext.

Question 985335: Three of the exterior angles of an n-sided polygon are 15, 25, 70, and the remaining exterior angles are 50 each. Find the value of n.
Answer by Edwin McCravy(13211)   (Show Source):
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The exterior angles of any polygon have sum 360°.

3 of them are 15°, 25° and 70° and the other (n-3) exterior angles are
50° each.

Sum of exterior angles = 15°+25°+70°+(n-3)*50° = 360°
110°+(n-3)*50° = 360°
(n-3)*50° = 250°

n-3 = 5
n = 8

So the polygon has 8 sides.

Edwin

Question 985194: One of the interior angles of the polygon with n sides is 124 degrees and the other (n-1) interior angles are each equal to 142 degrees. Find the value of n.
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the sum of the interior angles of a polygon of n sides is 180(n - 2).
124+(n-1)*142= 180(n-2)
124+142n-142 =180n-180*2
142n+124-142 =180n-360
142 n-18=180n-360
142n-18+360 =180n
142 n+342=180n
342=180 n-142 n
342=38 n
n=
9
Result : 9

Question 984961: 150n=180(n-2)
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150n = 180(n-2),

150n = 180n - 2*180,

2*180 = 180n - 150n,

360 = 30n,

n = = 12.

Question 984723: K=2
Dilation

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Do you have a question?

Question 983785: find scale factor and the value of x,y,and z.quad Abcd congruent to quad EFgh

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Right, I'll just get out my crystal ball, tarot cards, and chicken bones so that I can peer into the mystic mists and discern the diagram you are working from. Please, when posting questions here, use your head for something besides a hat rack.

John

My calculator said it, I believe it, that settles it

Question 983593: The sum of the interior angle of a regular polygon is 16 rt angle s . determine the number of sides of the polygon
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(n-2)*180 degrees=sum=90*16=1440, because each right angle is 90 degrees;n=number of sides
180n-360=1440
180n=1800
n=10 sides

Question 983226: In the figure above, OP Is a radius of the circle, PX is a tangent of the circle at point P, and the area of triangle OPX is 12. What is the area of the circle?
a. 16Pi
b. 12Pi
c. 18Pi
d. 24Pi
http://152.46.13.240/MoodleContent/SATprep/Math%20Unit%207/Lesson%203%20Assignment/cb2.gif

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The triangle  OPX  is right-angled triangle because the radius drawn to the tangent point is perpendicular to the tangent.
Therefore the area of the triangle  OPX  is half-product of its legs  OP  and  PX:

S = 12 = .|OP|.|OP| = *|OP|* = 3*|OP|.

It gives for the radius of the circle   r = |OP| = = 4.

Hence,  the area of the circle is   = = .

Answer.  The area of the circle is  .

Question 983192: The difference between the exterior angles of two regular polygon having sides equal to n and n+1 is 12. Find the value of n
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An exterior angle of a n sided polygon is equal to and that of sided polygon is .
if the difference between the exterior angles of two regular polygon is , we have
...........solve for

solutions:
if =>
if=>...=.> since cannot be negative number, disregard solution

Question 982337: Interior angle of regular polygon is 3 times its exterior.
a)find the size of exterior angle
b)the sum of interior angles
c)what do we Call such polygon

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The sum of the exterior angles of any polygon is 360 degrees. If each interior angle is 3 times the measure of an exterior angle, then the sum of the interior angles must be 3 times the sum of the exterior angles. I'll let you figure out how much that is.

Once you know the sum of the interior angles, use the formula for the sum of the interior angles

Sum of interior angles

Set this equal to the sum of the interior angles that you figured by multiplying 360 times 3, and then solve the equation for n, which is the number of sides of your polygon.

I have a polygon just like this, and I call mine George. You can look up the name of a polygon that has the number of sides you calculated.

John

My calculator said it, I believe it, that settles it

Question 982289: a student who was given a pentagon with four angle measures was asked to find the fifth angle the student said he would use [(n-2)times 180]/n. will his method work?
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That method might or might not work, depending on the pentagon. If not a regular pentagon, it can still be split into separate triangles.

Pick one vertex. Connect this with segments to the two non-adascent vertices. This will form THREE triangles.
180 degrees per triangle
3 triangles

The division by n WILL NOT WORK as a way to find the degrees per angle UNLESS this is a regular polygon. The above calculation is for 540 TOTAL degrees for the interior angles of the pentagon.

Question 982117: The two vertices that form the non-congruent side of an isosceles triangle are (-5,3) and (2,3). What are the coordinates of the other vertex.
I am beyond lost. Thanks!

Found 3 solutions by Edwin McCravy, macston, josgarithmetic:
Answer by Edwin McCravy(13211)   (Show Source):
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The first tutor gave one possible solution, but there are infinitely many possible answers.

(-1.5,5), (-1.5,7), (-1.5,0), (-1.5,100), (-1.5,-100), etc.

the x-coordinate can only be -1.5, but the y-coordinate can be any number
except the two values that produce an equilateral triangle, since there are
no non-congruent sides to an equilateral triangle.  Here are 4 solutions.
The last one is the one the first tutor gave.

Edwin

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The other vertex that forms the triangle is on a line perpendicular to the segment at its midpoint.
.
Find the midpoint:
midpoint=((x1+x2)/2,((y1+y2)/2)
midpoint=((-5+2)/2,(3+3)/2)
midpoint=(-(3/2),3)
.
Find the slope of the original segment:
m=slope
m=(y2-y1)/x2-x1
m=(3-3)/(2-(-5)=0/7=0
The line is horizontal.
.
The perpendicular line will be vertical, thus will have an undefined slope.
.
Equation for original line:
y=mx+b
y=3
.
The vertical line through the midpoint is x=-(3/2)
Any point on the line x=-(3/2) except (-(3/2),3) can be connected to the given vertices to form an isosceles triangle.

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Look at the points on a cartesian coordinate system. They show the endpoints of the base of the isosceles triangle. The other vertex would be on the y-axis and would be the center of a circle. Notice that the given points form the segment of the triangle parallel to the x-axis. This makes identifying the point on the x-axis to be easy. Look for the midpoint of x coordinates of -5 and 2.

The vertex is ( -3/2, 0 ).

Question 980709: a polygon has n sides. Two of its exterior angles are 90 degrees and 60 degrees. The remaining Exterior angles are each 14 degrees. Calculate the value of n
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a polygon has n sides. Two of its exterior angles are 90 degrees and 60 degrees. The remaining Exterior angles are each 14 degrees. Calculate the value of n
------------
The sum of exterior angles = 360 degs
360 - (90+60) = 210
----
210/14 = 15 angles
-->n = 17 sides

Question 980595: Need to find the length of the long side of a pentagon which has three 132 degree angles and two 72 degree angles and the four given sides are all 65.08'
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.
On top of that pentagon, let me draw a few lines, and my favorite figures: right triangles.
.
The long (horizontal) side at the bottom of that pentagon, ,
, , , and ,
with and .
So .
From the right triangles on the left side, I get
(rounded) and (rounded).
Then,

Question 980527: The areas of two similar polygons are 147cm squared and 75 cm squared. What is the ratio of their perimeters?
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The areas of two similar polygons are 147cm squared and 75 cm squared. What is the ratio of their perimeters?
--------------
A1:A2 = 147/75 = 49/25
P1:P2 = 7/5

Question 980345: A polygon has n sides
twp of the exterior angles are 41 each and five of the interior angles are 147 each
the remaining interior angles are 123.5 each.
Find the value of N

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Exterior angles are supplementary to the corresponding interior angle so there must be two interior angles that measure 139. That, together with the fact that there are five 147 degree interior angles means that there are interior angles that measure 123.5.

Since the sum of the interior angles in any polygon is given by , solve

for

John

My calculator said it, I believe it, that settles it

Question 979974: A regular polygon first maps directly onto itself after rotating 10 degrees. How many sides does the polygon have?
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A regular polygon first maps directly onto itself after rotating 10 degrees. How many sides does the polygon have?
--------
360/10 = 36 sides

Question 979574: :whus7h<

Question 979379: A hexagon has two sides each of length 3x inches. It has three sides each of length 2 inches. The sixth side has a length of 15 inches. If the perimeter of the hexagon is 135 inches, what is the value of x?
Answer by Edwin McCravy(13211)   (Show Source):
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A hexagon has two sides each of length 3x inches.
1st side = 3x
2nd side = 3x


It has three sides each of length 2 inches.
3rd side = 2
4th side = 2
5th side = 2


The sixth side has a length of 15 inches.
6th side = 15


If the perimeter of the hexagon is 135 inches,


3x + 3x + 2 + 2 + 2 + 15 = 135


what is the value of x?
Solve the equation and that will be the value of x.

Edwin

Question 979314: The ratio of the interior angle to the exterior angle of a polygon is 5:2.how many sides has the polygon
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The ratio of the interior angle to the exterior angle of a polygon is 5:2.how many sides has the polygon
--------------
5x + 2x = 180
x = 180/7
----------
Ext angles = 2x = 360/7
# of sides = 360/Ext
# of sides = 7

Question 979233: The following polygons are given. All of the polygons are
regular polygons.
Polygon a. Convex 15-gon
Polygon b. Convex 16-gon
Polygon c. Convex 17-gon
Polygon d. Convex 18-gon
Polygon e. Convex 19-gon
Polygon f. Convex 43-gon
Polygon g. Convex 44-gon
Polygon h. Convex 45-gon
Polygon i. Convex 46-gon
Polygon j. Convex 47-gon
1. Which polygon(s) has (have) interior angles that are whole
numbers (a number that is not a fraction or a decimal)? Explain
why it is that way.
2. What happens to the value of the interior angles as the
3. What happens to the value of the exterior angles as the
4. Explain what happens to the total sum of interior angles
as the number of sides in the polygon changes?
5. Explain what happens to the total sum of exterior angles
as the number of sides in the polygon changes?

Found 2 solutions by solver91311, Edwin McCravy:
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If is the number of sides and , then the measure of each interior angle is an integer.

As increases, the measure of the interior angles increases because gets closer to 180 as gets larger. A circle is the limiting shape as increases without bound so a circle is an infinite-sided polygon with an infinite number of vertices with 180 degree interior angles. It was this idea that allowed Archimedes to approximate by sandwiching a circle between two 96-sided polygons, one inscribed and the other circumscribed.

As increases, the measure of the exterior angles decreases because gets smaller as gets larger.

As increases, the total of the measures of the interior angles increases because gets larger as gets larger.

As increases, the total of the measures of the exterior angles remains constant because does not change as gets larger. Another way to put it is, no matter how many increasingly smaller turns you make, you still only go around one circle when you get back to where you started.

John

My calculator said it, I believe it, that settles it

Answer by Edwin McCravy(13211)   (Show Source):
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The following polygons are given. All of the polygons are
regular polygons.
Polygon a. Convex 15-gon
Polygon b. Convex 16-gon
Polygon c. Convex 17-gon
Polygon d. Convex 18-gon
Polygon e. Convex 19-gon
Polygon f. Convex 43-gon
Polygon g. Convex 44-gon
Polygon h. Convex 45-gon
Polygon i. Convex 46-gon
Polygon j. Convex 47-gon
1. Which polygon(s) has (have) interior angles that are whole
numbers (a number that is not a fraction or a decimal)? Explain
why it is that way.
The sum of the interior angles of a polygon of n-sides is

Since the polygons are regular, all the interior angles are the same,
so each one is that expression divided by n

That must be equal to a whole number, say, W. Since n does not divide
evenly into n-2, it must divide evenly into 180°. So we go through
the list to see which numbers divide evenly into 180°:

Polygon a. Convex 15-gon, yes, since 15 divides evenly into 180°.
Polygon b. Convex 16-gon, no
Polygon c. Convex 17-gon, no
Polygon d. Convex 18-gon, yes, since 18 divides evenly into 180°.
Polygon e. Convex 19-gon, no
Polygon f. Convex 43-gon, no
Polygon g. Convex 44-gon, no
Polygon h. Convex 45-gon, yes, since 45 divides evenly into 180°.
Polygon i. Convex 46-gon, no
Polygon j. Convex 47-gon, no


2. What happens to the value of the interior angles as the


So the value of the interior angles approaches 180° as the number
of sides of the polygon increases.


3. What happens to the value of the exterior angles as the
The sum of the exterior angles of any polygon is 360°.  So
each one of a regular polygon is

So the value of the exterior angles approaches 0° as the number
of sides of the polygon increases.


4. Explain what happens to the total sum of interior angles
as the number of sides in the polygon changes?



5. Explain what happens to the total sum of exterior angles
as the number of sides in the polygon changes?


Edwin`

Question 978878: each of the 5 angles is equal to 172 degree and the other angle is 160 each the number of sides of the polygon is
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each of the 5 angles is equal to 172 degree and the other angle is 160 each the number of sides of the polygon is
-------------
5*172 = 860 degs
860 + (n-5)*160 = (n-2)*180
860 + 160n - 800 = 180n - 360
420 = 20n
n = 21 sides
-------------
5*172 + 16*160 = 3420
180*(21 - 2) = 3420

can 35 degree be the exterior angle of a regular polygon

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n = 360/E

n = 360/35

n = 10.2857142857142

The result 10.2857142857142 is NOT a whole number. We cannot have a fractional number of sides.

So a 35 degree exterior angle is NOT possible for a regular polygon.

Question 978793: A regular polygon first maps directly onto itself after rotating 20 degrees. How many sides does the polygon have?
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sides, since

Question 978550: If a regular polygon has an angle of rotational symmetry that is 40° how many sides does the polygon have

Question 977524: The sum of the measures of the angles of any quadrilateral is 360°. The measures of ∠A and ∠B are the same. The measure of ∠C is 17° greater than the measure of ∠A, and the measure of ∠D is 37° less than ∠B. Find the measure of ∠A, ∠B, ∠C, and ∠D.
Found 2 solutions by solver91311, josgarithmetic:
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Solve for then calculate and

John

My calculator said it, I believe it, that settles it

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That is the system. Do you need more help?
-Yes!-

Try to use the first equation to eliminate A.

-
,
and notice how two of these equations are in terms of just B...
substitute for them in the "360" equation:

Work with this to solve for the value of B.
Do what you know you need to finish.

Question 976917: If the length of side a is 6 centimeters, the length of side b is 4 centimeters, and the length of side c is 7 centimeters, what is the measure of ∠B? Round your answer to two decimal places.
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This a triangle I take it.
Side a = 6cm
Side b = 4cm
Side c = 7cm
Find angle B
Using Cosine Rule
Cos B = (a^2 + c^2 - b^2)/ 2ac
Cos B = (6^2 + 7^2 - 4^2)/ 2 x 6 x 7
Cos B = (36 + 49 - 16)/ 84
Cos B = 69/84
Cos B = 0.8214
Angle B = 34.77 degrees (2 decimal places)
Hope this helps :-).

Question 976667: The interior angle of a regular polygon is thrice the exterior angle .how many sides has the polygon
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The sum of the interior and exterior angles must be 180 deg.
Let x = the measure of the exterior angle
Then x + 3x = 180 -> 4x = 180 -> x = 45
The exterior angles of a regular polygon add up to 360, which means each exterior angle is 360/n [n=number of sides]
So we have 360/n = 45 -> n = 360/45 = 8
So the polygon is an octagon (8 sides)

Question 976283: Here is my question:
A rhombus ABCD has a perimeter of 32cm and a diagonal AC of length 8cm. What is the exact length of diagonal BD?
Thank you very much! :)