# Unit 6 - Lakeside School

```Name __________________
Date
__________________
Honors Geometry Unit 6 Test
Circles
Find the indicated lengths and angles. Show calculations because partial credit will be
awarded for reasonable progress, but no detailed annotations or proofs are necessary.
Note: figures not necessarily drawn to scale.
a. Given: FD  9 , ED  3 , CB  5 , BA  10 , BA is a tangent
Find: CD and DA
E
F
D
C
B
A
b. Given: ABCD is an inscribed quadrilateral, with BC  CD and BC AD .
PQ is tangent to the circle at A. mBAQ  32 .
Find:
c. Given:
B , with radius 5 and the tangent segment CD  12 , and DE  FE .
Find: Length of DE and the LENGTH of FE (the arc length).
d. A half-circle is inscribed within an equilateral triangle such that the diameter of the halfcircle is centered on one edge of the triangle and the arc is tangent to the other two sides.
What is the diameter of the semi-circle if the triangle has an edge length of 4?
2. SHORT JUSTIFICATION
You do not need to write formal proofs, but your explanations should make it clear that you
understand the underlying geometric ideas. Brief explanations and/or annotated calculations
are sufficient.
a. PB is tangent to the circle at point B. Briefly explain how you can find the measure of
angle x WITHOUT using the secant-tangent angle theorem.
B
160
35
x
P
C
A
b. The steps and diagrams below show you how to construct a tangent to a circle from an
arbitrary point outside the circle. Briefly explain why the segment drawn in the final step
is guaranteed to be tangent to the original circle.
Given
C and point P outside the circle.
1. Construct segment PC.
2. Construct M, the midpoint of PC.
3. Construct M, centered at M with radius CM.
4. Place point I at one intersection of C and M.
5. Construct IP.
P
C
P
C
M
I
C
P
M
c. Circle C has a radius of R and x and y denote the lengths of the two parts of the chord.
Briefly explain why x  y  R 2  d 2 . Hint – you probably need to add some segments to
the diagram. Do not spend too much time on this one. It is challenging to see, so it will
be weighted less.
x
y
d
C
R
3. PROOF (JUSTIFICATIONS)
F
C
In the diagram to the right, BF and BE are tangents
to A and AC and AD are tangents to B .
The “eyeball theorem” states that given the
information above, MN  PQ .
M
P
B
A
Q
N
D
While the entire proof is too long to do here, we’ll
have you do pieces of it below.
E
a. Given the information above, prove that mAFB  mACB  90 and FAC  FBC
(call this angle 2 ).
F
C
M
P
B
A
N
Q
D
E
b. Using what you proved in part a (or just assuming a to be true), prove
mMFP  mMCP   .
F
C
M
P
B
A
N
Q
D
E
c. It can be shown that FCPM is an inscribed quadrilateral. Use this (and the results in a
and b if needed) to prove that mCMP  mCAB . Call this angle  .
F
C
M
P
B
A
N
Q
D
E
d. It can also be shown that ABCF is an inscribed quadrilateral. Use this (along with the
results in a, b, or c if needed) to prove that BAC  BFC   .
F
C
M
P
B
A
N
Q
D
E
Note that from here utilizing corresponding angles, CMP and CAB , it can be shown that
MP // AB // NQ . Moreover we can show that MN  AB and PQ  AB , and therefore MPQN
is a rectangle and consequently MN  PQ as we originally set out to prove. We’ll leave the
details to you as a fun, optional homework challenge should you choose to tackle it.
```