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EDEXCEL A-Level Maths
Pure Maths Year 1
Chapter 6 - Circles
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Created by
Sophia Lethbridge
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Cards (29)
How can you find the
midpoint
of a line segment?
By averaging the
x- and y-coordinates
of its
endpoints
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What is a
line segment
?
A finite part of a
straight line
with two
distinct endpoints
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What is the
perpendicular bisector
of a line segment AB?
It is the straight line that is perpendicular to AB and passes through the
midpoint
of AB
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What defines a
circle
in
geometry
?
A circle is the set of points that are
equidistant
from a
fixed point
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How can you derive equations of circles on a coordinate grid?
By using
Pythagoras' theorem
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What is the equation of a circle with center (0, 0) and radius r?
x² + y² = r²
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What is the general form of the equation of a circle with center (a, b) and radius r?
(x - a)² + (y - b)² = r²
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How can a straight line interact with a circle?
A straight line can
intersect
a circle
once
,
twice
, or not at all
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What is a
tangent
to a
circle
?
A straight line that intersects the circle at only one point
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What is the relationship between a
tangent
and the
radius
of a circle at the point of intersection?
A tangent is
perpendicular
to the radius at the point of intersection
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What is a
chord
in a circle?
A line segment that joins two points on the
circumference
of a circle
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What does the
perpendicular bisector
of a chord do?
It goes through the
center
of the circle
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What is a
triangle
and its properties related to circles?
A triangle consists of three points called
vertices
.
A unique circle can be drawn through the three vertices, called the
circumcircle
.
The center of the circumcircle is the
circumcentre
, where the
perpendicular bisectors
of each side intersect.
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What is the relationship between the
hypotenuse
of a right-angled triangle and its
circumcircle
?
The hypotenuse is a
diameter
of the circumcircle
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How can you state the result regarding a right angle in a
semicircle
?
If
∠PRQ
=
90°
, then R lies on the circle with diameter PQ
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What is the angle in a
semicircle
property?
The angle in a semicircle is always a
right angle
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How do you find the
center
of a circle given three points on the
circumference
?
Find the equations of the
perpendicular bisectors
of two different
chords
.
Find the coordinates of the point of intersection of the perpendicular bisectors.
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What is the summary of the
perpendicular bisector
property?
The perpendicular bisector of a line segment AB is perpendicular to AB and passes through its
midpoint
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What is the summary of the equation of a circle with center (0, 0)?
The equation is x² + y² = r²
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What is the summary of the general form of the equation of a
circle
?
The equation is x² + y² + 2fx + 2gy + c = 0
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What is the summary of the
tangent
property in relation to circles?
A tangent is
perpendicular
to the
radius
at the point of intersection
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What is the summary of the
chord
property in relation to circles?
The
perpendicular bisector
of a chord goes through the
center
of the circle
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What is the summary of the
circumcircle
property of triangles?
A unique circle can be drawn through the three vertices of any triangle, called the circumcircle
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What is the summary of the
right-angled triangle
property in relation to
circumcircles
?
The hypotenuse of a right-angled triangle is a
diameter
of the circumcircle
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What is the summary of finding the
center
of a circle using three points?
Find the equations of the
perpendicular bisectors
of two different
chords
and their intersection
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What is the
formula
for the
midpoint
of a line segment with endpoints
(
x
1
,
y
1
)
(x_1,y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2,y_2)
(
x
2
,
y
2
)
?
(
x
1
+
x
2
2
,
y
1
+
y
2
2
)
\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)
(
2
x
1
+
x
2
,
2
y
1
+
y
2
)
If the
gradient
of line segment AB is
m
m
m
, what is the gradient of its
perpendicular
bisector
?
−
1
m
-\frac{1}{m}
−
m
1
How can the
equation
of a
circle
also be expressed in the form
x
2
+
x² +
x
2
+
y
2
+
y² +
y
2
+
2
f
x
+
2fx +
2
f
x
+
2
g
y
+
2gy +
2
g
y
+
c
=
c =
c
=
0
0
0
?
This circle has
center
(-f, -g) and
radius
f
2
+
g
2
−
c
\sqrt{f² + g² - c}
f
2
+
g
2
−
c
What is the summary of the
midpoint formula
?
The midpoint of a line segment with
endpoints
(
x
1
,
y
1
)
(x_1, y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2, y_2)
(
x
2
,
y
2
)
is
(
x
1
+
x
2
2
,
y
1
+
y
2
2
)
\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
(
2
x
1
+
x
2
,
2
y
1
+
y
2
)
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