GCSE Circle Theorems

Июль 24, 2022

Содержание

2. Sector (Minor) Segment Diameter Radius Tangent Chord (Minor) Arc Circumference ? ? ? ? ? ?
3. Circle Theorems are laws that apply to both angles and lengths when circles are involved. We’ll
4. ? radius tangent “Angle between radius and tangent is 90°”. “Angle in semicircle is 90°.” Note
5. ? “Angles in same segment are equal.” “Angle at centre is twice the angle at the
6. ? Opposite angles of cyclic quadrilateral add up to 180.
7. Lengths of the tangents from a point to the circle are equal.
8. Radius is of constant length Bro Tip: When you have multiple radii, put a mark on
9. Identify which circle theorems you could use to solve each question. O 160° 100° ? Angle
10. Identify which circle theorems you could use to solve each question. 70° 60° 70° ? Angle
11. Identify which circle theorems you could use to solve each question. 115° ? Angle in semicircle
12. Identify which circle theorems you could use to solve each question. 70° ? Angle in semicircle
13. Identify which circle theorems you could use to solve each question. Angle in semicircle is 90°
14. Identify which circle theorems you could use to solve each question. Angle in semicircle is 90°
15. ? ?
16. 1 b c a d e f ? ? ? ? ? ?
17. 2 a b c d e ? ? ? ? ?
18. 3 a b c d ? ? ? ? ?
19. 4 ? ? ?
20. ☠ a b c ? ? ?
21. This one is probably the hardest to remember and a particular favourite in the Intermediate/Senior Maths
22. z = 58° ?
23. Angle ABC = Give a reason: Angle AOC = Give a reason: Angle CAE = Give
24. 1 2 3 ? ? ? ? ?
25. 4 5 6 ? ? ?
26. 7 8 ☠1 ? ? ?
27. ☠2 ?
28. A B C O a a 180-2a 2a 90-a 90-a Let angle BAO be a. Triangle
29. ? x a b b a ? ? Opposite angles of cyclic quadrilateral add up to
31. Скачать презентацию

Слайд 2

Sector
(Minor)
Segment
Diameter
Radius
Tangent
Chord
(Minor) Arc
Circumference
?
?
?
?
?
?
?
?
?

Слайд 3

Circle Theorems are laws that apply to both angles and lengths

when circles are involved. We’ll deal with them in groups.

#1 Non-Circle Theorems

These are not circle theorems, but are useful in questions involving circle theorems.

50

130

?

Angles in a quadrilateral add up to 360.

Base angles of an isosceles triangle are equal.

Слайд 4

?
radius
tangent
“Angle between radius and tangent is 90°”.
“Angle in semicircle is 90°.”
Note

that the hypotenuse of the triangle MUST be the diameter.

Bro Tip: Remember the wording in the black boxes, because you’re often required to justify in words a particular angle in an exam.

Слайд 5

?
“Angles in same segment are equal.”
“Angle at centre is twice the

angle at the circumference.”

Слайд 6

?
Opposite angles of cyclic quadrilateral add up to 180.

Слайд 7

Lengths of the tangents from a point to the circle are

equal.

Слайд 8

Radius is of constant length
Bro Tip: When you have multiple radii,

put a mark on each of them to remind yourself they’re the same length.

This result is that any triangle with one vertex at the centre, and the other two on the circumference, must be isosceles.

Слайд 9

Identify which circle theorems you could use to solve each question.
O
160°
100°
?
Angle

in semicircle is 90°

Angle between tangent and radius is 90°

Opposite angles of cyclic quadrilateral add to 180°

Angles in same segment are equal

Angle at centre is twice angle at circumference

Lengths of the tangents from a point to the circle are equal

Base angles of isosceles triangle are equal.

Angles of quadrilateral add to 360°

Reveal

Слайд 10

Identify which circle theorems you could use to solve each question.
70°
60°
70°
?
Angle

in semicircle is 90°

Angle between tangent and radius is 90°

Opposite angles of cyclic quadrilateral add to 180°

Angles in same segment are equal

Angle at centre is twice angle at circumference

Lengths of the tangents from a point to the circle are equal

Base angles of isosceles triangle are equal.

Angles of quadrilateral add to 360°

Reveal

Слайд 11

Identify which circle theorems you could use to solve each question.
115°
?
Angle

in semicircle is 90°

Angle between tangent and radius is 90°

Opposite angles of cyclic quadrilateral add to 180°

Angles in same segment are equal

Angle at centre is twice angle at circumference

Lengths of the tangents from a point to the circle are equal

Base angles of isosceles triangle are equal.

Angles of quadrilateral add to 360°

Reveal

Слайд 12

Identify which circle theorems you could use to solve each question.
70°
?
Angle

in semicircle is 90°

Angle between tangent and radius is 90°

Opposite angles of cyclic quadrilateral add to 180°

Angles in same segment are equal

Angle at centre is twice angle at circumference

Lengths of the tangents from a point to the circle are equal

Base angles of isosceles triangle are equal.

Angles of quadrilateral add to 360°

Reveal

Слайд 13

Identify which circle theorems you could use to solve each question.
Angle

in semicircle is 90°

Angle between tangent and radius is 90°

Opposite angles of cyclic quadrilateral add to 180°

Angles in same segment are equal

Angle at centre is twice angle at circumference

Lengths of the tangents from a point to the circle are equal

32°

?

Base angles of isosceles triangle are equal.

Angles of quadrilateral add to 360°

Reveal

Слайд 14

Identify which circle theorems you could use to solve each question.
Angle

in semicircle is 90°

Angle between tangent and radius is 90°

Opposite angles of cyclic quadrilateral add to 180°

Angles in same segment are equal

Angle at centre is twice angle at circumference

Lengths of the tangents from a point to the circle are equal

31°

?

Base angles of isosceles triangle are equal.

Angles of quadrilateral add to 360°

Reveal

Слайд 15

?
?

Слайд 16

1
b
c
a
d
e
f

?
?
?
?
?
?

Слайд 17

2
a
b
c
d
e

?
?
?
?
?

Слайд 18

3
a
b

c

d

?
?
?
?
?

Слайд 19

4

?
?
?

Слайд 20

☠
a
b
c

?
?
?

Слайд 21

This one is probably the hardest to remember and a particular

favourite in the Intermediate/Senior Maths Challenges.

?

The angle between the tangent and a chord...

Click to Start Bromanimation

...is equal to the angle in the alternate segment

Слайд 22

z = 58°
?

Слайд 23

Angle ABC =
Give a reason:
Angle AOC =
Give a reason:
Angle

CAE =

Give a reason:

112°

Supplementary angles of cyclic quadrilateral add up to 180.

136°

68°

Angle at centre is double angle at circumference.

Alternate Segment Theorem.

?

?

?

?

?

?

Source: IGCSE Jan 2014 (R)

Слайд 24

1
2
3

?
?
?
?
?

Слайд 25

4
5
6

?
?
?

Слайд 26

7
8
☠1

?
?
?

Слайд 27

☠2

?

Слайд 28

A
B
C
O
a
a
180-2a
2a
90-a
90-a
Let angle BAO be a. Triangle ABO is isosceles so ABO

= a. Remaining angle in triangle must be 180-2a. Thus BOC = 2a. Since triangle BOC is isosceles, angle BOC = OCB = 90 – a. Thus angle ABC = ABO + OBC = a + 90 – a = 90.

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?

?

?

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Слайд 29

?
x
a
b
b
a
?
?
Opposite angles of cyclic quadrilateral add up to 180.
This combined angle
=

180 – a – b
(angles in a triangle)

?

Adding opposite angles:
a + b + 180 – a – b = 180