Math 3395
Koppelman
HW # 17
Name__________________________
1. Last year, I received the following email from a former Math 3395 student at KSU:
Hello Mr. Koppelman, I am a former student of yours and I have a geometry
question for you. I have been playing around with Sketchpad and noticed
that if I inscribe a right triangle within a circle with the right angle being at
the center of the circle, the ratio of the arc directly opposite of the hypotenuse
to the hypotenuse is exactly 1.111. Why!!!?????
Using Geometer’s Sketchpad, verify the student’s conjecture. Respond to his question
(“Why!!!?????”).
2. Using Geometer’s Sketchpad, construct a circle and a random
point P outside the circle (as shown). Construct a line through P
that is tangent to the circle. Your line should remain a tangent
line no matter where point P is moved outside the circle.
P
Please email your construction to ckoppelm@kennesaw.edu
by noon on the day of our next class.
F
P
A
3. In the diagram, PF is a tangent segment to circle G. If PF = 15,
and AB = PC = 16, find the lengths of AP and DC .
B
C
G
D
C
4. In the circle, chord ̅̅̅̅
CD is parallel to diameter ̅̅̅̅
AB. By how many
degrees does the measure of angle C exceed the measure of angle D?
Feel free to use Geometer’s Sketchpad, but you should still be able to
prove that your answer is correct.
5. Concentric circles are circles with the same center. Let A be the
center of two concentric circles, as shown. PQ is a line segment
tangent to the smaller circle at point R and intersecting the larger
circle at points P and Q. If PQ = 20, what is the area of the region
lying between the two circles?
D
A
P
B
R
A
Q
A
6. In the diagram, all the sides of right triangle ABC are
tangent to the circle shown. If AB = 8 and BC = 15,
what is the radius of the circle?
C
B
P
7. In the diagram, two circles of radii 1 and 2 are tangent to
the x-axis at the origin. A segment that is tangent to the smaller
circle is drawn from point P, the y-intercept of the larger circle,
to point Q on the x-axis. If the coordinates of point Q are (a,0),
compute the value of a .
Q(a,0)
B
8. In the diagram, radii PA and PB are perpendicular and are 6 inches long.
Chord BC intersects PA at point D. If BD is three times as long as DC ,
A
find the length of chord BC to the nearest tenth of an inch.
D
P
C
9. Two circles are said to be tangent to each other if they
are each tangent to the same line at the same point.
For example, the circles in the diagram for question 7
above are each tangent to the x-axis at the origin, so
they are tangent to each other. They are said to be
internally tangent circles (because one is inside the other).
Circles can also be externally tangent, as in the accompanying
diagram.
Prove each of the following about externally tangent circles.
a) The line connecting the centers of two externally tangent circles passes through
their common point (their point of tangency).
b) In the diagram at the right, circles A and B are externally
tangent. Line PQ is tangent to circle A at point P and to
circle B at point Q. Prove that M is the midpoint of PQ .
c) In the diagram, if segments AM and BM are drawn,
prove that AMB is a right angle.
P
M
Q
A
B
Extra: If the radius of circle A is 5 and the radius of circle B is 3,
what is the length of PQ (answer to the nearest tenth).
Problems 10, 11, and 12 are optional. They are challenging.
A
10. In the diagram, line PQ is parallel to side BC of
triangle ABC, and intersects sides AB and AC
at P and Q, respectively. The circle passing through P
and tangent to AC at Q intersects AB again at R.
Explain why quadrilateral RQBC must be a cyclic
quadrilateral.
P
Q
R
C
B
C
11. In the diagram, triangle ABC is inscribed in circle P, with P
in the interior of the triangle. The measure of A is 54 degrees,
and the measure of B is 41 degrees. Find the measure of the
angle formed by radius PC and the altitude from C to AB (PCD).
A
(Hint: Extend radius PC into a diameter.)
12. In the diagram, radius PA has length 7 inches and the length of
segment BC is 8 inches. The measures of angles APB and PBC
are each 60 degrees. There are two possible values for the length
of segment PB. Find both. (Hint: Extend BP so that it becomes
a diameter.)
P
B
D
A
B
7
8
P
C
7