Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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G.G.51: Arcs Determined by Angles: Investigate theorems about the arcs
determined by angles intersecting a circle when the vertex is on the circle
1 In the accompanying diagram,
tangent to circle O at B, and
Find:
,
,
,
2 In the accompanying diagram,
.
Find
,
,
,
is inscribed in circle O,
bisects
,
is
,
is a diameter of circle O,
, and
and
are secants,
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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3 In the accompanying diagram, B is the midpoint of
O, chords
and
Find:
,
,
4 In circle O,
and
Find
,
intersect at E,
.
,
is a tangent,
.
,
,
, triangle ADC is inscribed in circle
is a tangent to circle O at D,
is a secant, and
,
is a diameter,
, and
is a secant,
is a radius,
,
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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5 Quadrilateral ABCD is inscribed in circle O,
secants,
, and
.
Find
,
,
,
,
,
are diameters,
and
are
, and
6 Given: circle O with
tangent to circle O at B, and
Find:
and
, diameter
.
,
,
is extended to C,
is
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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7 In the accompanying diagram of circle O,
intersect at E,
, chord
is drawn,
Find:
,
,
,
8 In the accompanying diagram,
diameter
at point E,
Find
,
,
,
,
and
are secants, chords
, and
.
.
is a tangent to circle O at point A, secant
, and
.
, and
and
intersects
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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9 In the accompanying diagram,
is tangent to circle O at D,
Find:
,
,
,
is isosceles with
is a secant, and chords
,
,
,
intersect at E.
,
10 In the accompanying diagram,
and intersects diameter
at G,
Find:
,
and
,
is inscribed in circle O. Secant
is a secant,
, and
,
bisects
.
,
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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11 In the accompanying diagram of circle O,
, and
.
Find:
,
,
,
is a secant,
,
12 In the accompanying diagram of circle O,
and
intersect at F, tangent
intersects circle O at C,
Find:
,
,
is tangent to circle O at T,
,
,
are secants, chords
,
, and
and
.
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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13 In the accompanying diagram of circle O,
and
is extended to C,
is a secant,
, and
Find
,
,
,
are chords,
is a diameter and
.
, and
14 In the accompanying diagram,
is tangent to circle O at D,
is a secant, chords
and
intersect at E, chord
is drawn,
,
is twice
, and
.
Find
,
,
,
and
.
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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15 In the accompanying diagram,
,
and
are tangent to circle O,
, and
.
Find
,
,
and
Is ABCD a parallelogram? [Explain your answer.]
16 In circle O,
is a tangent,
, and
.
Find:
,
,
is a secant,
,
is drawn.
,
,
,
are chords,
,
,
17 In the accompanying diagram of circle O, the ratio
Chord
is extended to external point M, chords
Find:
and
,
and
is
.
intersect at D, and tangent
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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18 In the accompanying diagram, chords
drawn,
,
,
Find
,
,
,
and
and
, and
intersect at Q, secants
and
is tangent to the circle at T.
.
19 In the accompanying diagram of circle O, diameter
intersects chord
tangent;
is a secant, , , and
are chords;
; and
Find:
,
,
,
,
are
.
at F;
.
is a
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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20 In the accompanying diagram of circle O with inscribed isosceles triangle ABC,
,
is a tangent, and secant
intersects diameter
at E.
Find:
,
,
,
21 In the accompanying diagram,
diameter
at point E, chord
Find:
,
,
,
,
is tangent to circle O at point A, secant
is drawn,
, and
,
.
intersects
.
,
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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22 In the accompanying diagram, isosceles triangle ABC is inscribed in circle O, and vertex
angle BAC measures 40º. Tangent
, secant
, and diameters
and
are drawn.
Find:
,
,
,
,
.
23 In the accompanying diagram, regular pentagon ABCDE is inscribed in circle O, chords
and
intersect at F, chord
is extended to G, and tangent
is drawn.
Find:
,
,
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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24 In the accompanying diagram of circle O, chord
intersects
at F, tangent
is drawn,
Find:
,
,
,
,
;
;
,
,
, secant
.
25 In the accompanying diagram of circle O, diameter
diameter
, tangent
, and chords
,
,
.
Find:
is parallel to diameter
, and
.
.
is extended through C to point P;
are drawn;
; and
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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26 In the accompanying diagram of circle O,
drawn; chord
is parallel to chord ; chord
tangent
is drawn.
Find:
,
,
,
,
; diameter
and chord
is extended through C to D; and
.
27 In the accompanying diagram of circle O,
,
tangent, secant
intersects diameter
at F, and secant
Find
,
,
,
,
are
,
is drawn.
,
is a
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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28 In the accompanying diagram of circle O, diameter
, chord , and radius
are drawn; and
a Find:
b Is
,
,
,
.
acute, right, obtuse or equiangular? Explain your answer.
29 In the accompanying diagram of circle O, diameter
are drawn;
; and
.
Find:
is extended through E to C; tangent
.
,
,
,
,
.
, chord
, and secants
and
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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30 In the accompanying diagram of circle O, tangent
intersects diameter
at F,
, and
Find:
,
,
,
Find:
are drawn;
,
,
; and
,
are drawn, secant
,
31 In the accompanying diagram of circle O, secant
chord
and chord
.
,
, secant
, tangent
.
, and
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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32 In the accompanying diagram of circle O, secant
drawn; chords
and
Find:
,
,
, secant
intersect at E, tangent
.
,
;
;
,
,
,
, chord
, and chord
, tangent
.
, and secant
to the nearest degree.
34 In the accompanying diagram of circle O, chords
are drawn;
,
; and
Find:
are
intersects circle O at C, and
33 In the accompanying diagram of circle O, tangent , secant
are drawn;
;
;
; and
.
Find:
, and chord
,
,
,
, and
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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35 In the accompanying diagram of circle O,
is tangent to the circle at A;
diameter
intersects chord
at E; chords ,
, and
are drawn;
and
is 32 more than
.
Find:
;
;
;
;
is a secant;
;
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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1 ANS:
40, 60, 70, 50, 60
REF: 010436siii
2 ANS:
90, 45, 40, 95, 30
REF: 018539siii
3 ANS:
60, 60, 105, 45, 60
REF: 068542siii
4 ANS:
20, 80, 60, 100, 30
REF: 088641siii
5 ANS:
100, 90, 80, 30, 40
REF: 088538siii
6 ANS:
36, 18, 42, 126, 108
REF: 088940siii
7 ANS:
55, 30, 50, 80, 75
REF: 068939siii
8 ANS:
160, 80, 50, 50, 100
REF: 068640siii
9 ANS:
45, 90, 22º30’, 22º30’, 90
REF: 018736siii
10 ANS:
72, 32, 45, 99, 61
REF: 068741siii
11 ANS:
50, 25, 130, 25, 155
REF: 088742siii
12 ANS:
20, 55, 90, 40, 125
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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REF: 068840siii
13 ANS:
20, 30, 40, 40, 100
REF: 018937siii
14 ANS:
72, 48, 24, 84, 24
REF: 069437siii
15 ANS:
130, 65, 50, 115, no, because
REF: 089439siii
16 ANS:
120, 35, 95, 30, 55
REF: 019537siii
17 ANS:
150, 60, 45, 90, 105
REF: 069537siii
18 ANS:
60, 50, 20, 100, 40
REF: 089537siii
19 ANS:
80, 55, 40, 85, 35
REF: 019639siii
20 ANS:
75, 30, 135, 45, 30
REF: 069636siii
21 ANS:
160, 80, 50, 50, 100
REF: 089636siii
22 ANS:
80, 20, 140, 30, 110
REF: 069737siii
23 ANS:
72, 72, 36
REF: 089738siii
24 ANS:
is not congruent to
Regents Exam Questions G.G.51: Arcs Determined by Angles 7
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60, 80, 100, 50, 30
REF: 019839siii
25 ANS:
120, 30, 30, 80, 90
REF: 069837siii
26 ANS:
120, 60, 90, 30, 60
REF: 089842siii
27 ANS:
50, 84, 48, 37, 95
REF: 019937siii
28 ANS:
120, 30, 30, 120, right because
REF: 069939siii
29 ANS:
80, 150, 35, 90, 105
REF: 089937siii
30 ANS:
40, 90, 40, 130, 100
REF: 010239siii
31 ANS:
80, 120, 20, 75, 70
REF: 060240siii
32 ANS:
60, 80, 10, 130, 50
REF: 080242siii
33 ANS:
80, 40, 20
REF: 010336siii
34 ANS:
140, 90, 60, 25, 95
REF: 060336siii
35 ANS:
74, 53, 67, 104, 90
REF: 080338siii