Question 1190528: Solve the problem and show the solutions.
The measure of the angle formed by two tangents to a circle is 80 degrees. Find the measure of the intercepted arc.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
Here is the operative geometry principle:
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When two tangents or secants from an external point intersect a circle,
the measure of the angle between the tangents or secants is half the
difference between the measures of the two intercepted arcs
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In this problem with two tangents, the two tangents divide the whole circle into two arcs with arc lengths x and (360-x) degrees.
So the 80 degree measure of the given angle is half the difference between the measures of those two arcs:
Solve using basic algebra....
Answer by ikleyn(52832)  (Show Source):
You can put this solution on YOUR website! .
Solve the problem and show the solutions.
The measure of the angle formed by two tangents to a circle is 80 degrees.
Find the measure of the intercepted arc.
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Make a sketch.
In the sketch, find two right-angled triangles.
(As a reminder, the radius drawn to the tangent point is a perpendicular to the tangent line).
In these triangles, you are given that the acute angles at the vertex outside the circle are half of 80 degrees, each,
i.e. 40 degrees.
(As a remainder, the line connecting the point outside the circle with its center is the bisector
of the angle between the tangent lines to the circle from this point).
Therefore, two central angles at the center of the circle are 90 - 40 = 50 degrees each.
It implies that the central angle intercepting the arc is the sum 50 + 50 = 100 degrees.
ANSWER. The intercepted arc is 100 degrees.
Solved.
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For properties of tangent lines to a circle, which I used in the solution, see the lessons
- A tangent line to a circle is perpendicular to the radius drawn to the tangent point
- Tangent segments to a circle from a point outside the circle
in this site.
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