ABC is an isosceles triangle inscribed within a circle of radius 4,such that side BC passes through the center.What is the length of the arc segment AC?
To find the length of the arc segment AC, we first need to determine the measure of the central angle that subtends this arc. Since triangle ABC is isosceles with side BC passing through the center of the circle, angle BAC must be an angle inscribed in a semicircle, which means it is a right angle.
Now, let's label the center of the circle as O. Triangle ABC is isosceles, so we can draw a perpendicular from O to side AC, intersecting at point D. Therefore, AD is the perpendicular bisector of side BC.
Since triangle ABC is isosceles, we can now determine the measure of angle BOC by using the properties of isosceles triangles. Angle BOC is formed by two radii of the circle and is therefore a central angle.
In a circle, the measure of a central angle is equal to twice the measure of the inscribed angle that it subtends. In this case, angle BAC is a right angle, so angle BOC must have a measure of 2 * 90° = 180°.
Now, we need to calculate the length of the arc segment AC. The length of an arc in a circle is given by the formula:
Arc Length = (Angle / 360°) * (2 * π * r),
where Angle is the measure of the central angle in degrees and r is the radius of the circle.
In this case, the Angle is 180° and the radius (r) is given as 4. Plugging these values into the formula, we have:
Arc Length = (180° / 360°) * (2 * π * 4) = (1/2) * (2 * π * 4) = 4π.
Therefore, the length of the arc segment AC is 4π units.