A circle has a center at #(1 ,3 )# and passes through #(2 ,4 )#. What is the length of an arc covering #pi /4 # radians on the circle?

Question

A circle has a center at #(1  ,3  )# and passes through #(2     ,4   )#.  What is the length of an arc covering #pi  /4  # radians on the circle?

Maverick Smith · Accepted Answer

#s = (sqrt2pi)/4# Compute the radius: #r² = (2 - 1)² + (4 - 3)²# #r = sqrt(2)# The arc length ,s, is: #s = rtheta# #s = (sqrt2pi)/4#

Jayden Jackson · Answer

undefined To find the length of an arc covering $ \frac{\pi}{4} $ radians on the circle, you need to follow these steps:

1. Find the radius of the circle using the distance formula between the center and the given point on the circle.
2. Once you have the radius, use the formula for the length of an arc on a circle, which is given by $ s = r \cdot \theta $, where $ s $ is the arc length, $ r $ is the radius, and $ \theta $ is the angle in radians.
3. Substitute the values you have into the formula to find the length of the arc.

Let's calculate:

1. Calculate the radius using the distance formula:

$$ \text{radius} = \sqrt{(2 - 1)^2 + (4 - 3)^2} $$
$$ \text{radius} = \sqrt{1^2 + 1^2} $$
$$ \text{radius} = \sqrt{2} $$

2. Use the formula for the length of an arc:

$$ \text{arc length} = \text{radius} \times \text{angle in radians} $$
$$ \text{arc length} = \sqrt{2} \times \frac{\pi}{4} $$
$$ \text{arc length} = \frac{\sqrt{2}\pi}{4} $$

So, the length of the arc covering $ \frac{\pi}{4} $ radians on the circle is $ \frac{\sqrt{2}\pi}{4} $.

Jayden Jackson · Answer

undefined To find the length of an arc covering π/4 radians on the circle, we first need to determine the radius of the circle. We can use the distance formula to find the distance between the center of the circle (1, 3) and the point on the circle (2, 4), which lies on the circumference of the circle.

Using the distance formula:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

$$d = \sqrt{(2 - 1)^2 + (4 - 3)^2}$$
$$d = \sqrt{(1)^2 + (1)^2}$$
$$d = \sqrt{1 + 1}$$
$$d = \sqrt{2}$$

So, the radius of the circle is √2.

Now, to find the length of the arc covering π/4 radians, we use the formula for the arc length of a circle:

$$s = rθ$$

Where:
- s is the length of the arc,
- r is the radius of the circle, and
- θ is the angle subtended by the arc (in radians).

Plugging in the values:
$$s = (\sqrt{2})(\frac{π}{4})$$
$$s = \frac{\sqrt{2}π}{4}$$

Therefore, the length of the arc covering π/4 radians on the circle is $\frac{\sqrt{2}π}{4}$ units.