A circle has a center at #(7 ,9 )# and passes through #(4 ,3 )#. What is the length of an arc covering #pi /4 # radians on the circle?
I found
We can use the 2 points to find the radius of the circle (evaluating the sdistance between them):
The arc length
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To find the length of an arc covering ( \frac{\pi}{4} ) radians on the circle, you need to follow these steps:
 Calculate the radius of the circle using the distance formula between the center and the given point on the circle.
 Use the formula for the circumference of a circle to find the total circumference.
 Use the formula for arc length to find the length of the arc for ( \frac{\pi}{4} ) radians.
Let's proceed with the steps:

Calculate the radius using the distance formula: [ \text{radius} = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} ] Given: [ (x_1, y_1) = (7, 9) ] [ (x_2, y_2) = (4, 3) ] [ \text{radius} = \sqrt{(4  7)^2 + (3  9)^2} ] [ \text{radius} = \sqrt{(3)^2 + (6)^2} ] [ \text{radius} = \sqrt{9 + 36} ] [ \text{radius} = \sqrt{45} ] [ \text{radius} = 3\sqrt{5} ]

Calculate the total circumference: [ \text{circumference} = 2\pi \times \text{radius} ] [ \text{circumference} = 2\pi \times 3\sqrt{5} ] [ \text{circumference} = 6\pi\sqrt{5} ]

Calculate the length of the arc for ( \frac{\pi}{4} ) radians: [ \text{arc length} = \text{radius} \times \text{angle in radians} ] [ \text{arc length} = 3\sqrt{5} \times \frac{\pi}{4} ] [ \text{arc length} = \frac{3\pi\sqrt{5}}{4} ]
Therefore, the length of the arc covering ( \frac{\pi}{4} ) radians on the circle is ( \frac{3\pi\sqrt{5}}{4} ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 A triangle has corners at #(2 , 2 )#, #(1 ,3 )#, and #(6 ,4 )#. What is the radius of the triangle's inscribed circle?
 A triangle has corners at #(4 ,7 )#, #(1 ,3 )#, and #(6 ,5 )#. What is the area of the triangle's circumscribed circle?
 A circle has a center at #(3 ,0 )# and passes through #(0 ,1 )#. What is the length of an arc covering #(3pi ) /4 # radians on the circle?
 A circle's center is at #(2 ,1 )# and it passes through #(0 ,7 )#. What is the length of an arc covering #(5pi ) /12 # radians on the circle?
 A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #(pi)/12 #, and the triangle's area is #6 #. What is the area of the triangle's incircle?
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