two tangents to a circle form an angle of 105 degrees. find the length of the major arc if the radius if the circle is 10 inches
Draw a diagram. Label the points of tangency A and B, and the intersection of the tangents, label T. The center of the circle is O.
Then angle T = 105°
angle OTA = OTB = 52.5°
triangles OBT and OAT are right triangles, so
angle TOA = TOB = 37.5°
so, angle AOB = 75°, making the major arc 285°
The radius of the circle does not matter.
length of the arc ... 285/360 * 2 * π * 10
Nice catch. I was just figuring the angular measure of the arc, not its length.
I wondered why they gave the radius!
To find the length of the major arc, we first need to determine the measure of the central angle corresponding to the angle formed by the two tangents.
When two tangents intersect outside a circle, the angle formed is equal to half the difference of the intercepted arcs. In this case, the intercepted arc is the major arc.
Let's call the intercepted arc "x" inches. Therefore, the other arc intercepted by the tangents is (360 - x), as the sum of the intercepted arcs is equal to the total circumference of the circle, which is 360 degrees.
Since the formed angle is 105 degrees, it is equal to half the difference of the intercepted arcs:
105 = (1/2) * (x - (360 - x))
Now, we can solve this equation to find the value of x:
105 = (1/2) * (2x - 360)
210 = 2x - 360
2x = 570
x = 285
Thus, the major arc is 285 inches long.