Circles notes

• What do you recall about circles?
Circle

• It has a center point that is equidistance from all points on the
circle.
Circle

circle.
• The distance from the center
point to any point on the circle
is called the radius.
Circle

circle.
• The distance from the center
point to any point on the circle
is called the radius.
• The segment that connects 2
points on the circle and goes
through the center point is called
the diameter.
Circle

Think about it...
?
• If you know the center point location and a point on a
circle, how could you verify another point is or is not
on a circle?

Think about it...
?
circle, how could you verify another point is or is not
on a circle?
• Can you find the distance
between the 2 known points?

Think about it...
?
circle, how could you verify another point is or is not on a
circle?
• Can you find the distance
between the 2 known points?
• What is the relationship
between the unknown
point and each of the given
points?

Distance Formula
• Recall you can use the distance formula to find the
distance between 2 points on the coordinate plane.

Distance Formula
• Recall all radii (plural of radius) in a circle have the
same measure.

Distance Formula
• Recall all radii (plural of radius) in a circle have the
same measure.
• If you know the center point and the measure of the
radius, you can determine whether any point lies on
the circle or not using the distance formula.

Is the point on the circle?
• Suppose we have a circle centered at the origin and
contains the point (0, 4). Does the point (3, √7) lie on
the circle?
?

the circle?
• We can verify, or prove, this
is true or false in 2 steps.
?

the circle?
1) Find the radius measure.
?

contains the point (0, 4). Does the point (3, √7) lie on the
circle?
1) Find the radius measure.
2) Find the distance between
the center and point. Compare
the distance to the radius.
?

Find the Radius
• Use the distance formula to find the distance
between the center and known point on the
circle.
0,0( )
0,4( )

Find the Radius
circle.
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
0,4( )
x2,y2( )
x1,y1( )

Find the Radius
circle.
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
0,4( )
x2,y2( )
x1,y1( )d = 4 - 0( )2
+ 0 - 0( )2

Find the Radius
circle.
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
0,4( )
x2,y2( )
x1,y1( )d = 4 - 0( )2
+ 0 - 0( )2
= 4( )2
= 4

Find the Radius
circle.
• The radius measures 4 units.
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
0,4( )
x2,y2( )
x1,y1( )d = 4 - 0( )2
+ 0 - 0( )2
= 4( )2
= 4

• Now find the distance between the center and given point.
Find the Distance to Point
?
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
3, 7( )

?
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
3, 7( )
x2,y2( )
x1,y1( )
d = 7 - 0( )
2
+ 3- 0( )2

?
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
3, 7( )
x2,y2( )
x1,y1( )
d = 7 - 0( )
2
+ 3- 0( )2
= 7( )
2
+ 3( )2

?
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
3, 7( )
x2,y2( )
x1,y1( )
d = 7 - 0( )
2
+ 3- 0( )2
= 7( )
2
+ 3( )2
= 7 + 9

?
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
3, 7( )
x2,y2( )
x1,y1( )
d = 7 - 0( )
2
+ 3- 0( )2
= 7( )
2
+ 3( )2
= 7 + 9
= 16 = 4

• The distance measures 4 units,
which is the same as the radius.
?
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
3, 7( )
x2,y2( )
x1,y1( )
d = 7 - 0( )
2
+ 3- 0( )2
= 7( )
2
+ 3( )2
= 7 + 9
= 16 = 4

• The distance measures 4 units,
which is the same as the radius.
• Therefore, the point (3, √7) lies on the circle.
?
d = y2 - y1( )2
+ x2 - x1( )2
0,0( )
3, 7( )
x2,y2( )
x1,y1( )
d = 7 - 0( )
2
+ 3- 0( )2
= 7( )
2
+ 3( )2
= 7 + 9
= 16 = 4

You try...
?
• A circle is centered at the origin and contains
the point (-7, 0). Prove or disprove that the
point (5, 5) lies on the circle.

You try...
?
• A circle is centered at the origin and contains
the point (-7, 0). Prove or disprove that the
point (5, 5) lies on the circle.
• Did you disprove the point
lies on the circle?
See how on the next slide.

Here’s how to disprove.• Find the radius using the origin (0, 0) and (-7, 0).

d = y2 - y1( )2
+ x2 - x1( )2

d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2

d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2
= -7( )2
+ 0( )2

d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2
= -7( )2
+ 0( )2
= 49 = 7

Here’s how to disprove.• Find the radius using the origin (0, 0) and (-7, 0).• Find the distance betweenthe center (0, 0) and (5, 5).
d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2
= -7( )2
+ 0( )2
= 49 = 7

d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2
= -7( )2
+ 0( )2
= 49 = 7
d = 5 - 0( )2
+ 5 - 0( )2

d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2
= -7( )2
+ 0( )2
= 49 = 7
d = 5 - 0( )2
+ 5 - 0( )2
= 5( )2
+ 5( )2

d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2
= -7( )2
+ 0( )2
= 49 = 7
d = 5 - 0( )2
+ 5 - 0( )2
= 5( )2
+ 5( )2
= 25 + 25

d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2
= -7( )2
+ 0( )2
= 49 = 7
d = 5 - 0( )2
+ 5 - 0( )2
= 5( )2
+ 5( )2
= 25 + 25 = 50 = 5 2

• The distances are not the same. This disproves
the statement that (5, 5) lies on the circle.d = y2 - y1( )2
+ x2 - x1( )2
d = -7 - 0( )2
+ 0 - 0( )2
= -7( )2
+ 0( )2
= 49 = 7
d = 5 - 0( )2
+ 5 - 0( )2
= 5( )2
+ 5( )2
= 25 + 25 = 50 = 5 2

Recap to prove or disprove a
point lies on the circle.

• Use the distance formula to find the radius of the circle
between the center point and point known on the circle.

• Use the distance formula to find the distance between
the center point and point unknown if on the circle.

• Compare the distances.

➡ Same distances proves the point is on the circle.

➡ Same distances proves the point is on the circle.
➡ Different distances disproves the point is on the circle.

Circles notes

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Circles notes