Chapter 10 Flashcards
What is a tangent?
a line that intersects a circle at exactly one point
Theorem 12-1:
if a line is tangent to a circle then the line is…
perpendicular to the radius drawn to the point of tangency
Theorem 12-2:
if a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle…
then the line is tangent to the circle
Theorem 12-3:
The two segments tangent to the outside of a circle are…
congruent
What is a chord?
A segment whose endpoints are on a circle
What is a diameter?
A chord that goes through the center of a circle
What is an arc?
pieces of a circle
what is a minor arc?
an arc that is less than 180 degrees
what is a major arc?
an arc that is more than 180 degrees
what is a semicircle?
an arc that is exactly 180 degrees
what is a central angle?
an angle made up of two segments whose vertex is the center of the circle
Theorem 12-4:
within a circle or in congruent circles… (three things)
- Congruent central angles have congruent chords
- Congruent chords have congruent arcs
- Congruent arcs have congruent central angles
Theorem 12-5:
within a circle or congruent circles… (two things)
- Chords equidistant from the center are congruent
2. Congruent chords are equidistant from the center
Theorem 12-6:
In a circle, a diameter that is perpendicular to a chord…
bisects the chord and its arcs
Theorem 12-7:
In a circle, a diameter that bisects a chord (that is not a diameter)…
is perpendicular to the chord
Theorem 12-8:
In a circle, the perpendicular bisector of a chord contains…
the center of the circle
What is an inscribed angle?
an angle whose sides are chords and whose vertex is ON the circle
Theorem 12-9:
the measure of an inscribed angle is…
half the measure of its intercepted arc
Two inscribed angles that intercept the same arc are…
congruent
An angle inscribed in a semicircle is always…
a right angle
The opposite angles of a quadrilateral inscribed in a circle are…
supplementary
Theorem 12-10:
The measure of an angle formed by a tangent and a chord is…
half the measure of an intercepted arc
What is a secant?
a chord that extends outside the circle, and has two endpoints lying on the circle
Theorem 12-11:
the measure of an angle formed by two lines that… (two things, one is inside, other is outside)
- intersect INSIDE a circle is half the SUM of the measures of the intercepted arcs (but not the center)
- intersect OUTSIDE the circle is half the DIFFERENCE of the measures of the intercepted arcs
Trick for 2 tangent circles
angle + arc = 180
Theroem 12-12 (Part 1):
how do you find part of the length of two chords intersecting inside a circle?
the product of the pieces of one chord is equal to the product of the pieces of the other chord
Theorem 12-12 (Part 2):
two secants intersect outside a circle
(whole secant) (outside piece) = (whole secant) (outside piece)
Theorem 12-12 (Part 3):
a tangent and a secant intersect outside a circle
(whole secant) (outside piece) = (tangent)^2
Theorem 12-13:
what is the equation of a circle?
(x-h)^2 + (y-k)^2 = r^2
where r is the radius and (h,k) is the center
this is known as standard form
How do you complete the square and get a circle in Standard Form?
- Gather the x terms and y terms so they are next to each other in the equation. Get the constant on the other side.
- Look at the coefficients of the x term and the y term. Divide the coefficients by 2 and square them. Add those numbers to BOTH sides
- Factor the x-trinomial and the y-trinomial. Both must factor into the same parenthesis twice. ex: (x-4)^2
The circle is not un the correct form.
If there is a plus (+) in the equation it is a…
circle!
if there is a minus (-) the the equation it is a…
ellipse or hyperbola! don’t fall for those!
Theorem 10-9: Circumference of a Circle
the circumference of a circle is π times the diameter
C= πd OR 2πr
What is an Arc Length?
a fraction of the circle’s circumference
Theorem 10-10: Arc Length
the length of an arc of a circle is the product of the ratio of the measure of the arc and the circumference of the circle
length = n
—— x 2πr
360
what is “Exact Area”
the answer in terms of pi (π)
Theorem 10-11: Area of a Circle
A = πr^2
Theorem 10-12: Area of a Sector of a Circle
Sector Area = n
—— x πr^2
360
