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Inscribed Angle
From Latin: in "inside" + scribere "to write"
Definition: The angle subtended
at a point on the circle by two given points on the circle.
Try this Drag any orange dot. Note that when moving the point P, the inscribed angle is constant
while it is in the
major arc
formed by A,B.
(If there is no image below, see support page.)
Given two points A and B, lines from them to a third point P form the inscribed angle ∠APB.
As you drag the point P above, notice that the inscribed angle is constant. It only depends on the position of A and B.
The inscribed angle is only defined for points on the major arc (the longest path around the circle between the two given points).
In the figure above, if you drag P around into the shorter (minor) arc it will be undefined.
Formula for inscribed angle
If you know the length of the minor arc and radius, the inscribed angle is:
Arcs and Chords
The two points A and B can be isolated points, or they could be the end points of an
arc
or
chord.
When they are the end points of an arc, the angle is sometimes called the peripheral angle of the arc.
Central Angle
A similar concept is the central angle. This is the angle subtended at the center of the circle by the two given points.
See Central Angle definition
The central angle is always twice the inscribed angle. See Central Angle Theorem. Relationship to Thales' Theorem
Refer to the above figure. If the two points A,B form a diameter of the circle, the inscribed angle will be 90°, which is
Thales' Theorem.
You can verify this yourself by solving the formula above using an arc length of half the circumference of the circle.
You can also move the points A or B above until the inscribed angle is exactly 90°.
You will see that the points A and B are then diametrically opposite each other.
Other circle topicsGeneral
Angles in a circleArcs
(C) 2009 Copyright John Page
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