Power Theorems Project
(Chord

Chord)
You will discover the relationship between the lengths of the parts of the chords.
1.)
To find this, add auxiliary lines to create two figures you DO know how to work with
(try for triangles).
2.)
Where
can you find congruent or similar triangles?
3.)
How do you know they are congruent/similar?
4.)
What does this tell you about the side lengths: x, y, w, z?
5.)
If you were on the right track, you should have stated Theorem 12

14. Compare your
result with
p.621. If you discovered this, write out the formal proof. If you didn’t,
return to step 1 and rethink your work.
Power Theorems Project
(Secant

Secant)
You will discover the relationship between the lengths of the parts of the se
cants.
1.)
To find this, add auxiliary lines to create two figures you DO know how to work with
(try for triangles).
2.)
Where can you find congruent or similar triangles?
3.)
How do you know they are congruent/similar?
4.)
What does this te
ll you about the side lengths: x, y, w, z?
If you were on the right track, you should have stated Theorem 12

15. Compare your
result with p.622. If you discovered this, write out the formal proof. If you didn’t, return
to step 1 and rethink your work
.
Power Theorems Project
(Secant

Tangent)
You will discover the relationship between the lengths of the parts of the secant (
) and
the tangent (
).
1.)
To find this, add auxiliary lines to cre
ate two figures you DO know how to work with
(try for triangles).
2.)
Where can you find congruent or similar triangles? [Hint: You will need to apply
Theorem 12

11. You will be shown the proof of this theorem on Friday/Monday by
your pe
ers.]
3.)
How do you know they are congruent/similar?
4.)
What does this tell you about the side lengths: w, y, and z?
5.)
If you were on the right track, you should have stated Theorem 12

16. Compare your
result with p.622. If you discovered this, write
out the formal proof. If you didn’t,
return to step 1 and rethink your work.
Power Theorems Project
Theorem 12

11:
The measure of an angle formed by a tangent and a chord is half the
measure of its intercepted arc
That is, if
is tangent to the circle at point A, and
is a chord, then
.
This is NOT one of the Power Theorems, but it is a theorem needed for the proof of one of
the Power Theorems.
Much like the Inscri
bed Angle Theorem, this proof can be broken into three cases. Once you
prove Case 1, you can use it to prove the other cases are true. You will only have proven the
theorem once you show that it works for ALL possible cases.
is the diameter
the center is OUTSIDE
the center is INSIDE
of the angle
of the angle
A nice guide for these proofs is in your book: #12

4(3) on p.647
There is a typo on part a (Case 1). Instead of using the picture
in the book, use:
where D is a point anywhere on the boundary of the circle,
on the left semi

circle.
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