A circle has a center at #(7 ,6 )# and passes through #(2 ,1 )#. What is the length of an arc covering #pi/12# radians on the circle?

Question

A circle has a center at #(7   ,6  )# and passes through #(2     ,1    )#.  What is the length of an arc covering #pi/12# radians on the circle?

Mason Adams · Accepted Answer

1.8512 units

Using distance formula, solve for the distance between (7,6) and (2,1). That would be #sqrt((7-2)^2 +(6-1)^2# equal to #5sqrt2# units. Since, arc length #S = rtheta# where r is the radius and #theta# is in radians. Multiplying: #5sqrt2 xx pi/12 = 1.8512 units#

Avery Allen · Answer

undefined To find the length of an arc covering π/12 radians on the circle, you can use the formula for the arc length of a circle:

Arc Length = r * θ

Where:
r is the radius of the circle,
θ is the central angle in radians.

First, you need to find the radius of the circle using the given center and a point on the circle. Then, you can calculate the arc length using the given central angle.

The distance between the center (7, 6) and the point (2, 1) can be found using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Substitute the given points:

Distance = √((2 - 7)^2 + (1 - 6)^2)
         = √((-5)^2 + (-5)^2)
         = √(25 + 25)
         = √50

The radius of the circle is √50.

Now, you can use the formula for arc length:

Arc Length = r * θ

Substitute the values:

Arc Length = √50 * (π/12)

Calculate:

Arc Length ≈ (1.581) * (π/12)
           ≈ 0.4132 * π

So, the length of the arc covering π/12 radians on the circle is approximately 0.4132π.