Two chords intersect with the measures shown in the drawing.

What is the value of x?

Solution:

we know that the product of the lengths of one intersecting chord must be equal to the product of the lengths of the other, so...

Find x.

What is the value of x?

Solution:


TV is a diameter, and SU is perpendicular to TV. This means that x=8

Triangle ABC is inscribed in a circle. AB is 12, AC is 6 and BC is 6 square root 3. Find arc BC.

These measures tell us that the triangle is a 30-60-90 triangle with a hypotenuse opposite the side AB and since we can see that arc BC is the arc of least measure, it must be 60 degrees

Write an equation for each translation of x^2 + y^2 = r^2 with the given radius. Right 5 units, down 3 units; radius 8.

Write the equation of the circle that is tangent to all four lines. x= -7, x= 11, y= -2, y= 16

Radius:
(11+7)/2=(16+2)/2=9
Center:
x: (11-7)/2=2
y: (16-2)/2=7

Points (-7,2) and (1,6) are endpoints of the diameter of a circle. What is the length of the diameter .

Write the equation in standard form of the circle with the given properties. Center at the origin; r = square root of 7

origin => C(0,0) with  
so 

 

becomes 

 

:)

Find the equation of the tangent to the circle with centre: (2,-2) at (5,-2)

if it is tangent at (5,-2) then the line must be perpendicular to the diameter, so x=5

Find the equation of the tangent to the circle with centre: (3,-1) at (-1,1)

the slope between (-1,1) and (3,-1) is 

m=(-1-1)/(3-(-1))=-1/2 
which must be perpendicular to our line's slope. Therefore using 
 
we have 

Find the equation of the circle that is tangent to the line x-1=0 and x-6=0, and passes through the point (2,2).

since our two lines are parallel, the circle's diameter must be 6-1=5 (and so r=5/2) 
This means that the x-coordinate of the center must 1+5/2=7/2 
Next we can write: 
 
after solving we determine that k=0. 
therefore... 

Find an equation of the circle whose diameter has endpoints (-5, 5) and (3, -1)

Midpoint is the center so 

(x,y)=((-5+3)/2,(5+(-1))/2=(-1,2) 


distance
 
so r=5 

our circle: 




:)

ok so my question is calculate the perimeter of the inscribed square, given that the radius of the circle is 3 ft. I dont really understand how you would know the sides of the inscribed square.

Since the square is inscribed, it's diagonals are diameters of the circle and we can say 
 

For a square the relationship between a side and the hypotenuse is 
 
so that 
 
And the perimeter of a square is 

 




:)

you are 18 feet away from a circular water tank and you are 42 feet away from a tangent. What is the radius of the circle?

Let x be the radius of the water tank, then 

r=x+18+42 

r=x+60


:)

What is the area of a circle that is inscribed in a square with an area of 4?

 
so  
for the circle... 
 
so  

 
 

Write an equation of a circle that has a center (-1,-3) and is tangent to the line 3x+y= -1

L: 3x+y+1=0 and P: (-1,-3) 
The perpendicular distance from L to P is to be the radius: 

 

From the equation of a circle: 

 

:)

an air plug on a bike is 15 inches from the center of the bike. if the wheel is spun 315 degrees clockwise, how far does the air plug travel?

We can find distance along a circle if our measurements are in radians. So... 

 

If we multiply both sides by 315 and simplify we see that 315 degrees is 

 


Now using the arclength formula we compute: 

 
which is about 82.47 inches 


:)