# Questions on Geometry: Circles and their properties answered by real tutors!

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Question 994568: a chord is 6 cm long. it is 15 cm from the centre of a circle. What is the radius of the circle.
Found 2 solutions by MathTherapy, josgarithmetic:
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a chord is 6 cm long. it is 15 cm from the centre of a circle. What is the radius of the circle.
```The perpendicular bisector is drawn from the circle's center to the chord, which it bisects
We now draw 2 radii to each endpoint of the chord
We now have an isosceles triangle which consists of 2 right triangles, with 2 radii as its hypotenuses.
To find the radius, we use the pythagorean theorem, and we get: . This results in a radius value of

```

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Imagine and draw a circle with center at the origin of cartesian system. y-axis makes a perpendicular bisector with a chord, and each part of the chord is 3 cm. in length. Two points on this circle are (-3,15) and (3,15).

The question asks for the radius of the circle. How far is either of the specified (or found) points from the origina? That is the radius.
Use the Distance Formula.

Question 994217: A square has an area of 32cm2
What is the area of the largest circle that can fit in it ??

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if a square has an area of , then =>=>=>

The diameter of the circle equals to
then, the area of the largest circle that can fit in it is:

Question 989033: If y= mx + c is a tangents to the circle x^2 + y^2 =r^2,show that c = rsqrt(1 + m^2}}}. Hence, find the equations of the tangents to the circle x^2 + y^2 = 4 which pass through the points (0,+_6).
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Centre of is (0,0) and radius r.
Since y=mx+c (or say mx-y+c=0) is tangent to this circle, distance from centre(0,0) to this line is equal to radius
So,

=>
=> --------------part (i)
Let y=mx+c be tangents to circle x^2 + y^2 = 4
Since, it passes through (0,6) and (0,-6)
So,
6=0+c and -6 = 0+c
=> c=6 or -6
Now, using part(i) proof,
or
=> or (from both equations-same values of m)
So equation of tangents are
and and and
Since there are two external points, hence two tangents can be drawn from each point. So, we have got 4 equations of tangents.

Question 993955: the coordinates of a the center of a circle is (4,3).if the circle passes the origin what's the radius of the circle
Found 2 solutions by Alan3354, Boreal:
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the coordinates of a the center of a circle is (4,3).if the circle passes the origin what's the radius of the circle
-------------
It's the distance from (0,0) to (4,3)

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The center is at (4,3)
The radius can be determined by the fact that the point (0,0) is on the circle.
Distance formula or Pythagorean Theorem
sqrt (4^2+3^2)=sqrt (25)=5

Question 993620: find the circumference of a circle with the diameter being 8
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Hi there,
Circumference of a circle = Pi x diameter
Circumference = Pi x 8
Circumference = 25.13 units (2 decimal places)
Hope this helps :-)

Question 992576: A circle with center A and a radius of 5 is tangent to a larger circle with center B and a diameter three times that of the smaller circle. What is the distance from A to B?
Found 2 solutions by addingup, vleith:
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A has a radius of 5, and 5x2= a diameter of 10.
B has a diameter 3 times bigger, so 3x10= a diameter of 30 And 30/2= radius 15.
So, the distance of center A to center B, assuming that the circles are externally tangential, is 20. Look at the drawing I made you.

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circle A has a radius of 5 and a diameter of 10
Circle B has a diameter 3 times that of circle A. So B's diameter is 3*10 = 30.
That makes B's radius 30/2 = 15.
What is the length from B to A? 15 + 5 = 20

Question 992559: Please help me with this problem: Determine whether the equation x^2+y^2-4x+2y=9 represents a circle, a point, or no graph.
I know that if there's a graph it should represent a circle, but i'm having trouble factoring properly.
Thanks!

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Determine whether the equation x^2+y^2-4x+2y=9 represents a circle, a point, or no graph.
------------
Complete the square on the x and on the y terms::
x^2-4x+4 + y^2+2y+1) = 9+4+1
-----
(x-2)^2 + (y+1)^2 = 14
------
Circle with center at (2,-1) and radius = sqrt(14)
------------
Cheers,
Stan H.
-----------

Question 992303: QUESTION:In circle O, diameter AB is perpendicular to chord CD at E. If CD=20cm, BE=2cm and AE= 4x-2, find the length of the diameter.
Using the intersecting chord theorem I know how to solve it, however neither values of CE or DE are given and you can't have two variables in an equation so I'm confused. By long shot i thought of using trigonometry but i don't think it's right. Thanks for the help in advance x

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the length of your diameter is 4x-2 + 2 = 4x.

if CD is equal to 20, this means that CE and ED are both equal to 10 because a diameter cuts a chord perpendicular to it into 2 equal segments.

here's what you have:

AB = the diameter = 4x.
AO = a radius = 2x.
OB = a radius = 2x.

since EB is equal to 2, then OE must be equal to 2x-2.

CE is equal to 10
ED is equal to 10

OC is equal to 2x because OC is a radius.
OD is equal to 2x because OD is a radius.

you have a right triangle formed that is OCE.
in that triangle:
OC is the hypotenuse which is equal to 2x.
OE is one leg which is equal to 2x-2.
CE is the other leg which is equal to 10.

by pythagorus, OE^2 + CE^2 = OC^2 which becomes:
(2x-2)^2 + 10^2 = (2x)^2
simplify this equation to get:
4x^2 - 8x + 4 + 100 = 4x^2
subtract 4x^2 from both sides of the equation to get:
-8x + 4 + 100 = 0
add 8x to both sides of the equation to get:
4 + 100 = 8x
combine like terms to get:
104 = 8x
solve for x to get:
x = 13.

if x = 13, then:
OC = 2x = 26
OE = 2x-2 = 24
CE = 10

by pythagorus, 10^2 + 24^2 = 26^2
this becomes:
100 + 576 = 676 which becomes:
676 = 676
this confirms the solution is correct.

you are asked to find the length of the diameter.
the length of the diameter is 4x which is equal to 4 * 13 which is equal to 52.

the key to solving is to know the geometric relationship that a chord perpendicular to a diameter is cut in half.

you also needed to determine that the length of the diameters was 4x-2 + 2 which made it equal to 4x.

you also needed to know that the triangle formed had a hypotenuse that was equal to the radius.

Question 992282: Complete the equation of the circle centered at (3 , -7 ) with radius 4.
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.
For circle with center at (h,k) and radius=r:
.

Question 991779: Given the circumference of a circle is 60 pi. Find the length of a 225 degree sector of this circle
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The circumference of the circle covers the entire 360 degrees.
You only want the length for 225 degrees.
Set up a ratio.

Question 991385: Find the center and radius of the circle: x^2+y^2-4x+10y-3=0
Found 4 solutions by Timnewman, ikleyn, josgarithmetic, Fombitz:
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Hi dear,
Comparing
x²+y²-4x+10y-3=0 with the general form of equation of a circle;x²+y²+2gx+2fy+c=0
From the above,
2g=-4
g=-2,
also,
2f=10
f=5
The centre of the
circle is (2,-5).
Also,
r²=g²+f²-c
r²=2²+5²-(-3)
r²=4+25+3

HOPE THIS HELPS?

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.
Find the center and radius of the circle: x^2+y^2-4x+10y-3=0
-----------------------------------------------------------------

= ,

Transform to an equivalent expression by completing full squares:

= ,

= ,

= .

It is the circle of the radius  r = = with the center at the point  (2,-5).

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Complete the Squares and put into standard form.
See this lesson for Completing the Squares, and do for both the x and the y.

Circle Equation Standard Form is and the center is (h,k), and size of radius is r.

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Center : (,)
.

Question 990630: A circle is enclosed in a square. The diameter of the circle is 2x. The square has sides of length 3x. Find the area of the shaded region in terms of x. Keepy pi.
Thank you very much!

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hi dear,
Area of the square=3x(3x)
=9x²
also,
Area of circle
Area of the
=x²(9-pi)

Question 990600: The diameter of a circle has endpoints J(12,25) and K(-10,13). Find the circumstance and area the circle. Use the pie key on your calculator and round to the nearest tenth.
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Hi dear,
Use
Where y1=25,y2=13,x1=12,x2=-10
put the above in the formula and get diameter of the circle to be 25.01
=12.51
circumference=2*pi*12.51
=78.61
area=pi*12.51²
=941.72

Question 990516: a circle has a center at (8,2). The point (3,7)is in the circle. What is the area of the circle to the nearest tenth of a square unit?
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The radius, , of a circle is the distance between any point on the circumference, and the center of the circle.
In this case, the point (3,7) must be at a distance from point (8,2), and
.
The area of a circle with radius is ,
so the area of the circle in the problem is
(rounded to the nearest tenth of a square unit).

Question 990244: You have been asked to design a can shaped like a right circular cylinder with height h and radius r. Given that the can must hold exactly 170 cm^3, what values of h and r will minimise the total surface area (including the top and bottom faces)? Give your answers correct to 2 decimal places as a list [in brackets] of the form: [ h, r ]
for constants h (height), r (radius), in that order.
THANK YOU

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, , and .

s is for surface area of the can.

v formula gives

Find ds/dr and equate to 0, and solve this for r.

Actually, you are looking for ------Solve this for r.

Question 990255: A circle passes through the Point (8,7) and touches the y-axis at the Point (0,3). Find the equation of the circle. if the circle cuts the x-axis at D and E, find the equation of another circle which is DE as diameter.

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Unknown center of the first circle, (h,k).
Using Distance Formula

This circle touching, just touching, the y-axis at (0,3) means that the center is some point on the line y=3. This means, you can solve for h, because you know . This is based on knowing how the standard circle equation works.

Center of this first circle is therefore, (8,-3). The point (0,3) on the circle may be the convenient point to again use the Distance Formula, to find the radius of this circle.

This first circle equation is then, .
I have not finished to do the final question, but maybe you can.

Question 990029: O is the center of the circle and m
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Not enough info.

Question 989575: what is the value of cot 0 and sec -15π/7?
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what is the value of cot 0
cot = 1/tan
tan(0) = 0
cot = 1/0 --> infinity or undefined.
====================
and sec -15pi/7?
-15pi/7 = -2pi - pi/7
sec(-x) = sec(s) --> sec(pi/7)
sec(pi/7) = 1/cos(pi/7)
I would use a calculator.

Question 987868: http://postimg.org/image/6omaybq21/
I tried approximating angle measures and ended up getting the right answer, J. However, I feel this is incorrect and would like to know a better way of solving it.

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use:
Inscribed angle theorem.
The measure of an inscribed angle is equal to one-half the measure of its intercepted arc.
so, angle °
now, in right angle triangle , if you connect and, the other two angles will be ° each
then we have triangle where one angle is ° and other two angles are and
if so, then the sum of all angles in a triangle is:

°

You upgraded the tires on your truck. The manufacturer tires are P245/70R16. You want upgraded to P285/75R16.
What is the diameter of the old tires?
What is the diameter of the new tires?
What is the circumference of the old tires?
What is the circumference of the new tires:

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You provide the specs of the tires.
----------
The diameter of the old tires or the new tires is not a math problem.
Look it up somewhere and post it, or go away.

Question 987660: Hi again!!
A curve is traced by a point P(x,y) which moves such that its distance from the point A(-1,1) is three times its distance from the point B(2,-1). Determine the equation of the curve.
IF you can figure this out, congratulations. You can do college math and I can't. :) Due Tuesday.

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The Distance Formula is the key to starting this solution.

Distance PA is Distance PB*3

Simplify this equation and convert into any meaningful form through your arithmetic/algebra skills.

Question 987537: John wants to construct a fence around his farm.
The farm is circular in shape with a radius of 1 ft.
What is the length of fencing material he will need to fence one complete circle around his farm?

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The farm is circular with a 1 foot radius? Small Farm! The farm must be maybe two potted plants.

feet of fence material.

You upgraded the tires on your truck. The manufacturer tires are P245/70R16. You want upgraded to P285/75R16.
What is the diameter of the old tires?
What is the diameter of the new tires?
What is the circumference of the old tires?
What is the circumference of the new tires:
If your odometer says you have traveled 15,000 miles, how far have you actually traveled with the new tires?
If your speedometer reads 70mph, can you get a ticket for going 5mph or more over the speed limit?

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Provide the specs for the tires.

Question 986902: Please help me solve this.The outside of a cylindrical structure at a factory must be painted.it radius is 3,5 m and its height is 8 m.How many litres of paint must be bought if 1 litre covers 10 m(2)?(the bottom of the structure will not be painted.)
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Hi there,
I assume the top of the cylinder will be painted
but, not the bottom.
Surface area of a cylinder
= pi* r^2 + 2* pi *r *h
= pi* 3.5^2 + 2 *pi *3.5* 8 (* times)
= 38.49 + 175.93
- 214.42 m^2
No. of litres required:
= 214.42/10
= 21.4 litres.
Hope this helps :-)

Question 986754: I really need help with this, would appreciate any help.
Show that the point E(-1 , 2) lies on the circumference of the circle with equation
x2 + y2 – 2x – 3y – 1 = 0 and find the coordinates of the point F,
given that EF is a diameter.

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True, so point E satisfies the equation and therefore lies on the circle.
.
.
.
Complete the square in x and y to find the center of the circle,

So the center of the circle is (,)
.
.
.
The x distance from E to the center is the same as the x distance from the center to F.

Similarly for y,

So point F is (,)
.
.
.
.

Question 986443: Points O(0,0) and B(0,3) below lie in the standard (x,y) coordinate plane. The collection of all points such that each is twice as far from B as from O forms a circle. The point (sqrt 3, 0) is 1 point on the circle. What are the coordinates of the center of the circle?
F. (sqrt 3/2, 3/2)
G. (0,3/2)
H. (0, 1)
J. (0,-1)
K. (0,-3)

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You are saying Distance between (x,y) and B(0,3) is 2 times distance between (x,y) and O(0,0). You are expecting, according to description, the points (x,y) are a circle.

Want standard form, requiring Completing the Square

Circle, centered at (0,-1), and radius is .

Question 986087: An arc of a circle of radius 7cm is 14cm long.what does d arc subtend at d centre of d circle

Found 2 solutions by Alan3354, josgarithmetic:
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An arc of a circle of radius 7cm is 14cm long.what does d arc subtend at d centre of d circle
--------------------
Angle = Arc/r = 14/7 = 2 radians

Question 985805: The angle subtended by a chord at the centre of a circle is
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90°

(because it leans the arc of  180°).
 Figure.  An inscribed angle leaning on the diameter is a right angle

See the lesson An inscribed angle in a circle in this site.

Question 985461: The points X(3,4) and Y(9,1) lie on the circumference of a circle. There is exactly 60 degree of arc between X and Y. Find the radius of the circle.
Found 2 solutions by MathTherapy, josgarithmetic:
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The points X(3,4) and Y(9,1) lie on the circumference of a circle. There is exactly 60 degree of arc between X and Y. Find the radius of the circle.
```Line segment XY =
With center O, radii OX and OY are congruent
With arc XY being , central angle XOY also equals
With radii OX and OY congruent, an equilateral triangle is formed
With line segment XY being , radii also measure:

```

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The center point of the circle is equally distant from both X and Y, forming an isosceles triangle; but since you have central angle being 60 degrees, this forces the other angles each to also be 60 degrees. This is then an equilateral triangle with points, X, Y, and the center of the circle. The radius will be the DISTANCE between X and Y, found using the distance formula. Now you know radius r.

Next, simply form and compute or evaluate the length of arc.
(Fraction of the circumference)

Question 985460: Points A, B, and C lie on the circumference of a circle. AB is twice the radius of the circle. Find m(angle)ACB.
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Solution

Since  AB  is twice the radius of the circle,  AB  is the diameter of the circle.

The angle  LACB  is  90°,  because it leans on the diameter.

Question 985245: The formula for the circumference of the basketball is c=2pi(r). If the circumference of a basketball is 29inches is it a regulation size basketball?

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SIZE BEST FOR CIRCUMFERENCE
7 Men's Official Men, boys 12 and older 29 1/2 - 30 in 75 - 76 cm
6 Women's Official Women, girls 7 and older, boys ages 7-12 28 1/2 - 29 in 72 - 74 cm
5 Youth Kids under 7 27 - 28 in 69 - 71 cm
3 Mini Miniature basketball best for small kids 22 - 22.5 in 56 - 57 cm

It is much easier to find it in GOOGLE than to print and to submit the huge text to this site.  Isn't?

Question 985135: 16. The circumference of the circular pool is 31.4m. The path around it is 1m wide. Find the circumference of the outer boundary of the path.
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.

=
.
The 1 meter path adds 2 meters to the diameter, so new diameter is 12 meters.
=

Question 985081: a circle has a center (8,2). the point (3,7) is on the circle. What is the area of the circle to the nearest tenth of a square unit?
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a circle has a center (8,2). the point (3,7) is on the circle. What is the area of the circle to the nearest tenth of a square unit?
---------------
Find the distance between the points. That's the radius r.

Question 985080: a circle has a center (3,5). the point (3,8) is on the circle. What is the circumference of the circle to the nearest unit?
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.
Radius is distance from (3,5) to (3,8) is 3 units.
Diameter=2r=2(3)=6 units
Circumference=(pi)(diameter)=19 units

Question 985052: Give the equation of the circle tangent to both axes, of radius 5 and in the 1st quadrant. In the 2nd Quadrant.In the 3rd quadrant. In the 4th quadrant.
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.
The distance from each axis to center of circle must equal radius for the circle to be tangent to both axes.
.
In the first quadrant, center of circle is +5 from each axis, center at (5,5)
.
In second quadrant, center of circle is +5 from y axis (x=5) and 5 below x axis (y=-5), center at (5,-5)
.
In third quadrant, center of circle is 5 left of y axis (x=-5) and 5 below x axis (y=-5), center at (-5,-5)
.
In fourth quadrant, center of circle is 5 left of y axis (x=-5), 5 above x axis (y=5), center at (-5,5)
.
Standard form of circle:

where (h,k) is the center and r is the radius
.

.

.

.

Question 985050: Find the equation of circle having (10,2) and (6,-4) as ends of diameter.
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Hi there
First find the mid point of the
diameter. This will give you the
centre of the circle.
Mid point = x1 + x2/2, y1 + y2/2
Mid point = 10 + 6/2 , 2 + (-4)/2
Mid point = 16/2 , -2/2
Mid point = 8 , -1
Distance from (8,-1) to (10,2)
Distance formula = √(x2 - x1)^2 + (y2 - y1)^2
Distance formuls = √(10 - 8)^2 + (2-(-1))^2
Distance formula = √(2)^2 + (3)^2
Distance formula = √ 4 + 9
Distance formula = √ 13
Circle formula:
(x - a)^2 + (y - b)^2 = r^2
Using Midpt (8, -1) and radius = √13
(x - 8)^2 + (y + 1)^2 = 13
Hope this helps :-)

Question 984845: A circle has a center (3,6) and radius 8. Find all values of y such that (-2, y) is a point on the circle.
may be more than one answer list smaller then larger
Provide exact expressions, not decimal approximations.