Mixed Review on Formulas & Theorems on
Geometry of Circles
Circle Formulas and Theorems :http://www.mathwarehouse.com/geometry/circle/
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(except questions 4 and 5 which were taken from jmap.org )
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6) UR and TQ are chords. M UT  52 o
What is the measure of QR,UV , TS , TQS , mURV ?
7) M QR = 80o
What is the measure of UAT , QBR , UT ?
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8) 1  2 and QR =70o
What is the measure of 1 , 2 , QOR
Is VS || QR? Explain your answer?
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Question 4 and 5 were taken from jmap.org . Included below are the worked out
answers provided from that excellent website.
4
GM and ML form a semi-circle and
Given circle O with diameter GOAL; measure 126° ( 7  180) and 54°
10
secants HUG and HTAM intersect at
3
point H; mGM : mML : mLT  7 : 3: 2; (  180) , respectively. LT measures
10
and chord GU  chord UT. Find the
36°. GM and ML form a semi-circle
ratio of mUGL to mH.
and measure 126°. GUT measures 144°
(180  36). Equal chords intercept equal
arcs. Because chord GU  chord UT,
144
GU and UT each measures 72° (
).
2
mUTL  108 (72  36). The measure of
an inscribed angle is half that of its
intercepted arc. So mUGL  54.
The angle formed by a tangent and a
secant is equal to half the difference
between the intercepted arcs:
126  72
 27
2
The ratio of mUGL to mH is 54:27,
or 2:1.
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5
BA, AG and GF form a semi-circle and
In the accompanying diagram, circle O
3
2
measure 90° (  180), 60° (  180)
has radius OD, diameter BOHF , secant
6
6
1
CBA, and chords DHG and BD; CE and 30° (  180) , respectively. The
6
is tangent to circle O at D; mDF  80; measure of an inscribed angle is half that
of its intercepted arc. So mBDG  75
and mBA : mAG : mGF  3: 2 :1.
Find mGF , mBHD, mBDG, ( 90  60 ) and mHBD  40 ( 80 ).
2
2
mGDE, mC , and mBOD.
Therefore mBHD  65 (180-(75+40)).
The angle formed by a tangent and a chord is
half the intercepted arc. Since the intercepted
arc is 110° (80+30), mGDE  55. Given
diameter
BOHF
and
mDF  80,
mBD  100. The angle formed by a
tangent and a secant is equal to half the
difference between the intercepted arcs,
so mC :
(60  30  80)  100
 35
2
The measure of a central angle is equal to
the measure of the arc it intercepts, so
mBOD  100.
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Mixed Review on Formulas & Theorems on
Geometry of Circles
Circle Formulas and Theorems :http://www.mathwarehouse.com/geometry/circle/
© www.mathwarehouse.com
(except questions 4 and 5 which were taken from jmap.org )
All Rights Reserved
Commercial Use Prohibited
TEACHERS: Feel free to make copies of this worksheet for the sole purpose of use in
your own classroom. ENJOY!!! Redistribution in any other form is prohibited.
More worksheets and activities available at
www.mathwarehouse.com/classroom/worksheets-and-activities.php
Play Math Games at TheMathGames.com
© www.mathwarehouse.com