Power Theorems Project

nostrilswelderElectronics - Devices

Oct 10, 2013 (3 years and 6 months ago)

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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.