Ch 10.4-10.6 Circles
The length of a secant and a tangent that meet outside a circle...
(Total)² = external segment × (total)
What are the differences between an inscribed angle and a central angle?
1. See how the position of their VERTICES B and O is not the same? While B is on the circle, O is at the center. 2. The measure of a central angle (or the intercepted arc) = TWICE its inscribed angle's measure.
When two nonparallel lines intersect outside a circle, the measure of the angle formed is...
1/2 the difference of the measures of the intercepted arcs.
If a tangent and a chord intersect at a point on a circle, then the measure of the tangent-chord angle is ...
1/2 the measure of its intercepted arc. (Tangent and Intersected Chord Theorem)
The measure of a circumscribed angle (the angle formed when two tangents meet outside) is equal to ...
180° - the central angle that intercepts the arc.
What is the name of an angle whose vertex is on a circle and whose sides are determined by two chords?
An inscribed angle.
The area of circle corresponds to a complete rotation of 360°. Yet, the area of a sector then corresponds to a central angle, which is fraction of a complete rotation:
Area of a sector = (ɵ°/360°)×πr², where ɵ is the central angle.
When two chords intersect inside a circle, the measure of the vertical angles formed is the average of their corresponding intercepted arcs:
Both angles x and E = 120° because (70°+170°)÷2 = 120°
Because about 3.14 × diameters are needed, one after the other, to complete one circle around, the formula for Circumference is then...
C= dπ or C=2πr
To convert radians to degrees
Degrees = radians ×180°/π
The length of two secant segments that meet outside a circle...
External segment × (total) = external segment × (total)
An arc that lies between two lines, rays, or segments is called an...
Intercepted arc. (to subtend = to connect, to hold)
When the length (S) of an intercepted arc is exactly the same as the length of the radius of the circle, the measure of such a central angle is one...
Radian in 1 radian, the arc length equals the radius. S = r
To convert degrees to radians
Radians = given degrees × π/180°
The length of an arc is a segment of the circumference: The total perimeter of a circle (the circumference) corresponds to one complete rotation, 360°. Similarly, the length of a segment of the circumference, an intercepted arc, will correspond to a smaller rotation ɵ°, (central angle°). Therefore the length of an arc, S, is going to be a fraction of the entire circumference:
S = (ɵ°/360°) × C where ɵ = central angle° and C = dπ
The length of an arc when the central angle, ɵ, is given in radians:
S = ɵ∙r
If 2 chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
Segments of chords theorem
If a polygon is drawn outside a circle, such that its sides are tangents, it is said to be...
circumscribed about the circle
If two inscribed angles intercept the same arc, then such inscribed angles will be...
congruent. (Inscribed angles of a circle theorem)
If a right triangle is inscribed in a circle, then the hypotenuse is a ...
diameter of the circle. (Inscribed right triangle theorem)
The opposite angles of an inscribed quadrilateral to a circle are...
supplementary. (Inscribed Quadrilateral Theorem)