A chord of a circle is a line segment whose endpoints lie on the circle. A secant of a circle is a line that intersects the circle at two points An inscribed angle is the angle formed by two chords having a common endpoint. The other endpoints define the intercepted arc.
Inscribed Polygons Inscribed Polygons A polygon is said to be inscribed in a circle if all its vertices are on the circumference of the circle. Not every polygon can be inscribed in a circle. For example, a parallelogram which is not a rectangle cannot be inscribed in a circle, because a circle containing three of its vertices cannot contain the fourth: A polynomial that can be inscribed in a circle is called a cyclic polynomial.
For example, the following pentagon is cyclic: Inscribed Triangles All triangles can be inscribed in a circle, and the center of the circle is the intersection of any two perpendicular bisectors of its sides.
This works because points on the perpendicular bisector of a segment are equidistant from its endpoints: Triangle ABC is inscribed in a circle Prove: The center of that circle is at the point of intersection of the perpendicular bisectors of AB and AC.
Let P be the intersection of those two perpendicular bisectors.
Likewise, every point on the perpendicular bisector of AC is equidistant from A and C, so P is also equidistant from these points: Had we chosen to use the perpendicular bisectors of AB and BC instead, we would get the same result, since there can only be one center of a circle.
So a consequence of this is that the perpendicular bisectors of the sides of any triangle must meet at a common point. The circle in which the triangle is inscribed is called its circumcircle, and we can also say that the circle has been circumscribed about the triangle. The center of that circle is called the circumcenter of the triangle.
An Application The idea of finding the circumcenter of a triangle using perpendicular bisectors of its sides has an important application in archaeology: While on an archaeological dig, Fred uncovered a piece of a circular dish.
He wanted to know how big the original dish was, so he needed to determine its diameter. To do this, he laid the piece on a coordinate plane with gridlines spaced 1 cm apart, and found three points on the dish's edge were at coordinates 0, 0—4, 2and 6, 2. These three points determine a triangle with those vertices, so he found the equations of two perpendicular bisectors of the sides, set their y-values equal to find their point of intersection, and then was able to find the radius as the hypotenuse of a right triangle from that point to the origin: If we set the y's of these two lines equal to each other, we have: So the point 1, 7 is center of the circle, and we can now find the radius since it is the distance from the center to any of the three vertices of the triangle.
The distance from 1, 7 to 0, 0 is or about 7. Therefore the diameter is about Conjecture 1: In a circle, the measure of an inscribed angle is half the measure of the central angle with the same intercepted arc. (A central angle is any angle whose vertex is located at the center of a circle.).
Write a conjecture about an angle whose vertex is on a circle and 1. inscribed angles that intercept the same arc are congruent. 2.
An angle inscribed in a semicircle is a right angle. Z 2 and the angle intercept the same arc. By Corollary 1, the angles are congruent, so m Z 2 If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. Since Ø A and Ø B both intercept arc CD, m Ø A.
The measure of an angle inscribed in a circle is one-half the measure of the central angle. Inscribed Angles Intercepting Arcs Conjecture Inscribed angles that intercept the same arc are congruent.
Write this measurement inside the circle, near 𝐷𝐷 𝐶𝐶. Inscribed angles that intercept the same arc are _____ Circle Conjecture #4 Angles inscribed in a semicircle are _____ angles. Circle Conjecture #1 The measure of a central angle is the _____as the measure of the arc it.
If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. 3. The measure of an inscribed angle is half the measure of its intercepted arc. 4.
An inscribed angle subtends a semicircle if and only if the angle is a Inscribed Angles Write paragraph proofs for Exercises 1 and 2.