Let be a point on the plane. Let's consider all the points on the plane whose distance to is equal to
The collection of all such points defines a geometrical shape which is called a circle.
The point is called the center of the circle.
The distance from to any point on the circle is called the radius.
In our diagram above, the radius is
Consider the diagram below. What is the measure of if
Notice that the points and lie on the circle. From the diagram, the radius of the circle is
To find we first notice that
Now, and are both segments between the center of the circle and the circle itself. So, they are both radii of the circle, and because the radius is we have
Finally, we obtain
What is the measure of $\overline{OC}$ if $AB = 8?$
|
a
|
$8$ |
|
b
|
$3$ |
|
c
|
$4$ |
|
d
|
$5$ |
|
e
|
$6$ |
What is the measure of $\overline{AB}$ if $OC=8?$
|
a
|
$8$ |
|
b
|
$18$ |
|
c
|
$16$ |
|
d
|
$14$ |
|
e
|
$4$ |
The diagram below shows a circle of radius a point that lies on the circle, and a third point on the plane.
Which of the following statements are true about the diagram above?
The point lies on the circle itself. So, the distance from to is equal to the radius of the circle:
On the other hand, the point lies strictly outside the circle. So, the distance between and the center of the circle must be greater than the radius:
Therefore, statements II and III are true, while statement I is false.
The diagram above shows a circle of radius $7$ and the points $C$ and $P$ that lie on the circle.
Which of the following statements are true?
- $OP > 7$
- $OC < 7$
- $OP = 7$
|
a
|
I and III only |
|
b
|
III only |
|
c
|
II only |
|
d
|
I only |
|
e
|
II and III only |
The diagram above shows a circle of radius $10,$ a point $C$ that lies on the circle, and a third point $P$ on the plane.
Which of the following statements are true?
- $OP > 10$
- $OP \leq 10$
- $OC = 10$
|
a
|
II and III only |
|
b
|
I and II only |
|
c
|
III only |
|
d
|
II only |
|
e
|
I and III only |
There are three important line segments associated with a circle:
A radius is any line segment that connects a point on the circle with the center (left-hand diagram).
A chord is any line segment whose endpoints lie on the circle (middle diagram).
A diameter is any chord that passes through the center of the circle (right-hand diagram). The diameter is the longest chord and its measure is twice the radius of the circle:
Which of the following statements are true about the diagram above?
- is a chord of the circle
- is a diameter of the circle
- is a radius of the circle
Let's analyze each statement in turn.
Statement I is true. As we can see, both endpoints and lie on the circle. Therefore, is a chord.
Statement II is true. As we can see, both endpoints and lie on the circle and passes through the center Therefore, is a diameter.
Statement III is true. The segment connects a point on the circle with its center. So, is a radius.
In conclusion, statements I, II, and III are true.
Which of the following statements are true about the diagram above?
- $\overline{OA}$ is a diameter of the circle
- $\overline{OB}$ is a radius of the circle
- $\overline{BC}$ is a chord of the circle
|
a
|
I and II only |
|
b
|
I and III only |
|
c
|
II and III only |
|
d
|
I, II, and III |
|
e
|
III only |
Which of the following statements are true about the diagram above?
- $\overline{AB}$ is a diameter of the circle
- $\overline{BC}$ is a radius of the circle
- $\overline{OA}$ is a chord of the circle
|
a
|
I and III only |
|
b
|
III only |
|
c
|
I only |
|
d
|
II only |
|
e
|
I and II only |