Math B Assignments: Introduction to Proofs
Geometry: Circle Geometry
|Day |Topics |Homework |HW Grade |Quiz Grade |
|1 |Circles, central angles and arcs |HW Circles - 1 | | |
|2 |Tangents and chords |HW Circles - 2 | | |
|3 |Inscribed angles |HW Circles - 3 | | |
|4 |Practice **QUIZ** |HW Circles - 4 | | |
|5 |Interior and exterior angles |HW Circles - 5 | | |
|6 |Practice |HW Circles - 6 | | |
|7 |Lengths of tangent segments and secant segments |HW Circles - 7 | | |
|8 |Practice **QUIZ** |HW Circles - 8 | | |
|9 |Quadrilaterals inscribed in circles |HW Circles - 9 | | |
|10 |Proofs |HW Circles - 10 | | |
|11 |Review **QUIZ** |HW Circles - Review | | |
|12 |***TEST*** | | | |
Answers to selected problems
HW - 1
1a. 40( b. 70( 2. 105( 3. 125( 5. 60( and 150( 6. 50(
7. [pic] 8. 90( 9a. 2 b. 76( c. 8
HW – 2
5. 34 6. 3 7a. 17 b. 9 c. 28.1( d. 61.9(
8a. (22.97 b. (25.46 9a. [pic] b. 26.7 c. 18( 10a. 17 b. 28.1(
11. 7.3 cm
HW – 3
2. 40( 3. 240( 4. 150( 5. 30( 6a. 110( b. 70( c. 120(
7a. 40( b. 70( c. 90( d. 70( 8. (2x)( 10. 25(
11a. 100( b. 70( c. 70(
HW – 4
3a. 16 b. the longer one
4a. [pic] b/c it is closer to the center of the circle.
b. [pic] b/c it is opposite the shortest side (farthest from center of circle)
5a. 4 common tangents b. 3 common tangents c. 1 common tangent
HW – 5
1. 25( 2. 120( 3. 40( 4a. 130( b. 140( c. 180 – x 5. 60(
6. 75( 7a. 40( b. 100( c. 150( d. 20( e. 50( f. 95(
7g. 145( h. 70( 8a. 130( b. 50( c. 60( d. 95( e. 125(
HW – 6
1a. 40( b. 40( c. 20( d. 140( e. 50(
2a. 85( b. 70( c. 37.5( d. 22.5( e. 77.5(
3a. 120( b. 80( c. 80( d. 60( e. 100( f. 30( g. 50(
4a. 130( b. 140( c. 40( d. 95( e. 85( f. 90( g. 45( h. 135(
HW – 7
2. 8 3. 14 4. 4 5. 5 6. 1 7. 5
8a. 21 b. 5 c. 2 d. 8
HW - 8
1e. [pic] or appx. 3.328 2a. 120 b. 36 3. 2
4a. 68( b. 88( c. 136( d. 2.25 5. 21
HW – 9
1a. 100(, 120(, 80(, and 60( b. 110(, 110(, 70(, and 70( c. Keep thinking.
3. 50(, 100(, 130(, and 80( 4. TV has no value. It is a vast wasteland. Except channel 12.
5a. 2.25 mi b. 3.75 mi c. 5.89 mi d. 106.26( e. 73.74( f. 2.41 mi
Review
1. 30( 2. 160( 3. 50( 4. 80( 5. 40( 6. 25( 7. 110(
8. 70( 9. 90( 10. 80( 11. 36( 12. 50( 13. 90( 14. 16
15. 3 16. 4 17. 5 18. 6.5 19a. 19 b. 135(
20a. 160( b. 10 21a. 13 b. 12 c. 67.4( d. 112.6(
22a. 80( b. 20( c. 120( d. 80( e. 10( f. 60( g. 140(
Geometry HW: Circles - 1
Name
1. Points A and B are on circle O. If [pic], find
a. m(AOB
b. m(OAB (you need to add [pic] to your diagram)
2. In circle O, [pic] and [pic] are diameters. If [pic], find [pic].
3. In circle O at right, [pic] is a diameter, [pic], [pic], and [pic]. Find the numerical value of [pic].
4. A chord is a line segment that joins two points on a circle.
a. We want to prove: Congruent central angles subtend (cut off) congruent chords.
Given: In circle O, (AOB ( (COD
Prove: [pic]
b. 1) What is the converse of the theorem from part a?
2) Explain briefly how the proof from part a could be modified to prove the converse of the theorem.
5. In circle O in the diagram at right, chords [pic] and [pic] are congruent, [pic] and [pic]. Find the measures of arcs [pic] and [pic].
6. In circle O at left, [pic] is a diameter and ABCD is
an isosceles trapezoid. If (AOD = 65(, find [pic].
7. Write the equation of the circle having center (1, –2) and radius 3.
8. In circle O at right, [pic], [pic], (AOF ( (DOE, m(FOE = 40( and [pic]=60(. Find m(AOF.
9. In circle O at left, [pic] is a diameter and (AOB ( (COD. [pic], [pic], [pic] and [pic].
a. Find the value of x.
b. Find [pic]
c. Find the value of y.
Geometry HW: Circles - 2
Name
1. We wish to find the center of a circle by construction.
a. Draw a chord on the left side of the circle
(do not make it too short, but do not try to
make it a diameter, either). Construct the
perpendicular bisector of this chord.
What have you constructed?
b. Find the center of the circle.
2. Construct a tangent to circle O at point A.
3. We want to prove the theorem from today’s notes: Two tangent segments
drawn to a circle from the same external point are congruent.
Given: Circle O with tangent segments [pic] and [pic];
[pic], [pic], [pic] are drawn.
Prove: [pic]
4. Based on your proof for #1 above,
a. Explain why [pic] bisects (APB.
b. Name two other things [pic] bisects.
In problems #3 and #4, (ABC is circumscribed about a circle with points of tangency at D, E and F. (Another way of saying this is that the circle is inscribed in the triangle.)
5. If AD = 2, BE = 5 and AC = 12, find the perimeter of (ABC.
6. If [pic], AF = 2 and FC = 10, find BD.
In problems #7 and #8, [pic] and [pic] are tangents to circle O and P, Q, O and R are collinear.
7. If PA = 15 and OA = 8, find the values of
a. OP b. PQ
c. m(APQ d. [pic]
8. If Q is the midpoint of RP and RP = 12, find
a. the perimeter of quadrilateral AOBP and
b. the area of quadrilateral AOBP.
9. A and B are points on circle O, [pic] and [pic].
a. Find the length of a radius of circle O.
b. Find the circumference of circle O to the nearest tenth.
c. Find the area of circle O in terms of (.
10. In circle O in the diagram at right, chord [pic] is perpendicular to diameter [pic]. If [pic]and [pic],
a. find the length of the radius of circle O.
b. find m(AOD to the nearest tenth of a degree.
11. A chord of length 9.6 cm is 5.5 cm away from the center of a circle. What is the radius of the circle?
Geometry HW: Circles - 3
Name
1. a. Construct a hexagon inscribed in
the circle (with center shown) at right.
b. Explain how we could change the
construction to create an inscribed
equilateral triangle.
2. Find the measure of an inscribed angle that
intercepts an arc of degree measure 80(.
3. Find the degree measure of the arc intercepted
by an inscribed angle of 120(
4. Triangle BAT is inscribed in a circle.
If [pic] and m(BAT = 40(, find [pic].
5. Isosceles triangle PIG with vertex angle P is
inscribed in a circle. If [pic] find the
measure of (IPG.
6. In the circle at right, [pic] is a diameter, m(BAE = 30( and [pic]. Find the values of
a. m(AEB
b. m(BEC
c. [pic]
7. In circle O at left, diameter[pic] is perpendicular to chord [pic], diameter [pic] is drawn, and [pic]. Find the values of
a. [pic] b. [pic]
c. m(CTW d. m(TWC
8. Acute (ABC is inscribed in circle O. If m(ABC = x(, express the measure of (AOC in terms of x.
9. Use the givens below to prove the following theorem (and remember it):
The arcs between two parallel chords of a circle are congruent.
Given. In a circle, chords [pic] and [pic] are drawn; [pic]
Prove: [pic] (Hint: Draw either [pic] or [pic].)
10. In circle O, chord [pic] is parallel to diameter [pic] and C is between A and D. If [pic], find [pic].
11. In circle O in the diagram at right, secants [pic] and [pic] are parallel, [pic] and [pic].
a. Find[pic]
b. Find [pic]
c. [pic]
Geometry HW: Circles - 4
Name
1. Prove the theorem from earlier in the unit: If a radius is perpendicular
to a chord, then it bisects the chord and its arc.
Given: Circle O with chord [pic], radius [pic]([pic], [pic] and [pic];
[pic] and [pic]are drawn.
Prove: a. [pic]
b. [pic]
2. Prove the theorem from earlier in the unit: If two cords are congruent,
they are equidistant from the center of the circle.
Given: Circle O with chords [pic]and [pic], [pic],
radius [pic]([pic], radius [pic]([pic];
[pic] and [pic]are drawn.
Prove: [pic]
3. A circle has radius 8.
a. What is the length of the longest possible chord that can be drawn in this circle?
b. Two chords in this circle have lengths 6 and 7. Which chord is closer to the center of the circle? Justify your answer.
4. Triangle ABC is inscribed in circle O. Chord [pic] is 2 units from O, chord [pic] is 5 units from O and chord [pic] is 6 units from O.
a. Which is the longest side of (ABC? Explain how you know.
b. Which is the smallest angle of (ABC? Explain how you know.
5. A common tangent is a line that is tangent to two different circles.
a. Circles P and Q do not intersect. Draw all the common tangents
to these two circles.
b. Circles R and S are externally tangent (they intersect
in one point but neither circle is inside the other).
Draw all the common tangents to these two circles.
c. Circles T and V are internally tangent (they intersect in
one point and one circle is inside the other). Draw all
the common tangents to these two circles.
Geometry HW Circles - 5
Name
For problems #1 and #2, use the diagram of the circle with secants
[pic] and [pic]. (The problems are separate.)
1. If [pic] = 80( and [pic] = 30(, find m(R.
2. If [pic]:[pic] = 3:2 and m(R = 20(, find [pic].
3. In the diagram at right, [pic] is a secant and [pic] is a tangent to the circle. If [pic] is 80( more than [pic], find m(P.
4. [pic] and [pic] are distinct tangents to a circle.
a. If [pic] = 50(, find m(P.
b. If m(P = 40(, find [pic].
c. If [pic] = x, find m(P in terms of x.
5. In a circle, chords [pic] and [pic] intersect at E .
If [pic] = 20( and [pic] = 100(, find m(AEC.
6. In a circle, chords [pic] and [pic] intersect at E. If m(BED = 100(
and [pic]:[pic] = 3:5, find [pic].
7. In the diagram at right. [pic] is tangent to circle O at A,
[pic] = 70(, m(P = 15(, and [pic]:[pic] = 3:2. Find
a. [pic] b. [pic]
c. [pic] d. m(PAB
e. m(BCD f. m(CED
g. m(ABP h. m(PAD
8. In the diagram at right, [pic] and [pic] are tangent to the circle and [pic] is a diameter. If [pic] = 120( and m(P = 50(, find
a. [pic] b. [pic]
c. [pic] d. m(DEC
e. m(PAC
9. Prove that the measure of an exterior angle of a circle is half the difference of the measures of the intercepted arcs.
Given: In circle O, [pic] and [pic] are secants.
Prove: [pic]
Add chord [pic], label the appropriate arcs and angles,
and do the proof algebraically like the one in class.
Geometry HW: Circles - 6
Name
1. In the diagram at right, [pic] is tangent to circle O at B, m(DOA = 100( and m(ABF = 20(. Find
a. m(CAB b. [pic]
c. m(E d. m(EAB
e. m(C
2. In the diagram at right, [pic]is tangent to circle O, [pic] = 45(,
m(BEC = 65(, [pic] = x, and [pic] = 2x + 20(. Find
a. [pic] b. [pic]
c. m(CPA d. m(BAC e. m(BAP
3. In the diagram at right, [pic] is tangent to
circle O at U, diameter [pic] is extended
to S, [pic] = 60(, [pic] = 40( and
[pic]:[pic] = 4:3. Find
a. [pic] b. m(ERA
c. [pic] d. [pic] e. m(ERN
f. m(BSE g. m(NBS
4. In the diagram at right, [pic] and [pic] are tangents to circle O and [pic] is a diameter. If [pic] = 50( and m(P = 40(, find
a. [pic] b. [pic]
c. [pic] d. m(RIC
e. m(RIN f. m(OCB
g. m(MRY h. m(MRP
Geometry HW: Circles - 7
Name
1. Prove that when two chords intersect inside a circle, the product of the parts of one chord is equal to the product of the parts of the other chord.
Given: In a circle, chords [pic] and [pic] intersect at E.
Prove: (AE)(BE) = (CE)(DE)
(Hint: Draw chords [pic] and [pic].)
2. In a circle, chords [pic] and [pic] intersect at E,
CE = 12, ED = 2 and AE = 3. Find BE.
3. In a circle, chords [pic] and [pic] intersect at E,
AE = 3, EB = 16 and CE:ED = 3:4. Find CD.
4. In a circle, [pic]is tangent to the circle at F, and
secant [pic] intersects the circle at B and C.
If AB = 2 and BC = 6, find AF.
5. In a circle, [pic]is tangent to the circle at F, and
secant [pic] intersects the circle at B and C.
If AF = 10 and BC = 15, find AB.
6. Line segments [pic] and [pic] are secants to a
circle from external point A. If AB = 3, BC = 7
and AD = 5, find DE.
7. In a circle, diameter [pic] is extended through B to external point P and tangent segment [pic] is drawn to point C on the circle. If BP = 4 and PC = 6, find the length of [pic].
8. In the diagram at right, [pic] and [pic] are secants to the circle; chords [pic] and [pic] intersect at E with BE > EC. If PA = 14, AB = 10, PC = 16, AE = ED = 4 and BC = 10, find
a. PD b. CD
c. CE d. EB
Geometry HW: Circles - 8
Name
1. a. On the grid at right, graph the circle having equation [pic]
(a compass would be a good idea).
b. On the same grid, graph the line lines
y = 1 and [pic]. Label the point where they intersect P; label the point where y = 1 is tangent to the circle T and label the points where [pic] intersects the circle S and R (from left to right toward P).
c. Find the length of [pic].
d. Find the length of [pic].
e. Find the length of [pic]. (Note: the coordinates of S are not (3, 5). Think of a method other than the distance formula.)
2. In the diagram of circle O at right (not to scale),
[pic] = 4:5:4:2.
a. Find m(BOC
b. Find m(BPC
3. In the diagram at right, a circle is inscribed in an equilateral triangle with sides of length [pic]. Find the radius of the circle. (Hint: draw radius [pic] and segment [pic]. Then figure out m(OAE.)
4. In the diagram at right, isosceles triangle ABC has vertex A on a circle and base [pic] tangent to the circle at T where T is the midpoint of [pic]. If BC = 12 and the perimeter is 44, find
a. m(C to the nearest degree.
b. [pic] to the nearest degree.
c. [pic] to the nearest degree.
d. BR
5. Circles R and Q have radii 3 and 4 respectively. They are externally tangent to each other and also both externally tangent to larger circle P. What must be the radius of circle P so that (QRP has a right angle at R? (Note: “externally tangent” means none of the circles is inside another.)
Geometry HW: Circles - 9
Name
1. An inscribed quadrilateral (a.k.a. a cyclic quadrilateral) is a quadrilateral whose four vertices all lie on the same circle.
a. Find the measures of each angle in ABCD b. Find the measures of each angle in FGHJ
c. State a property about pairs of angles that hold for both of the inscribed quadrilaterals above.
2. Complete the proof below that opposite angles in a quadrilateral inscribed in a circle are supplementary.
Given: Quadrilateral ABCD inscribed in a circle
Prove: (A and (C are supplementary.
3. Quadrilateral ABCD is inscribed in a circle, m(A = x,
m(B = y, m(C = y + 30 and m(D = x + y – 70. Find
the measures of all four angles.
4. Quadrilateral QRST is inscribed in a circle; its diagonals
intersect at V. If QV = 8, RV = 6 and SV = 9, what is the
value of TV?
5. The Senior Class of Nowhere High School is taking an overnight trip to Round Pond State Recreation Area. The bus stops first at the visitors center (V). From there, students have a choice. They can continue by bus 5 more miles to the Lumpy Bed Lodge (L) on the edge of Round Pond. Or, if they are feeling adventurous, they can hike four miles to the boat dock (D) and then paddle a canoe to the wilderness camping area (W) on the shore of the pond directly opposite the lodge.
a. How far will those who choose to camp have to paddle their canoes?
b. How far will the campers have to paddle to visit those staying at the lodge?
c. How far would someone at the lodge have to walk on Pond’s Edge Trail to visit people at the wilderness camp?
d. What is the degree measure of [pic]?
e. What is the degree measure of [pic]?
f. What is the length in miles of [pic]?
Geometry HW: Circles - 10
Name
1. In the accompanying diagram of circle O, diameter [pic] is drawn, tangent [pic]is drawn to the circle at B, E is a point on the circle, and [pic].
a. Prove: (ABE ~ (CAB
b. If AE = 6 and EB = 8, find the length of BC.
2. In the accompanying diagram, [pic] and [pic] is the diameter of circle O.
Prove: a. [pic]
b. [pic]
3. In the diagram, [pic] is tangent to circle O at A, [pic], [pic], [pic], [pic] bisects [pic].
a. Prove: 1) (OAN ~ (ORQ
2) (TAQ ~ (OAN
b. If OP = 30 and QS = 48, find the values of
1) RQ 2) RO
3) AN 4) ON
5) AT 6) QT
Geometry Review: Circles
Name Date
Find the value of x or y in each diagram. Note that the diagrams are not necessarily to scale.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18.
19. In circle O, chords [pic] and [pic] intersect at E.
a. If AE = 6, EB = 10, CE = x – 1 and DE = 3x,
find the numerical length of [pic].
b. If [pic] = 3y, [pic] = 4y, and [pic] = 30(,
find the numerical value of [pic].
20. In circle O, P is an external point, [pic] is a
secant segment and [pic] is a tangent segment.
a. If [pic] = 9x + 17, [pic] = 7x – 9, and m(P = 25(,
find the numerical value of [pic].
b. If TP = 5 and QR is 5 more than PQ,
find the numerical length of [pic].
21. A circle is inscribed in isosceles trapezoid ABCD with bases
AB = 8 and DC = 18.
a. Find the lengths of the legs of ABCD.
b. Find the length of the diameter of the circle.
c. Find m(C.
d. Find [pic].
22. In the diagram at right, [pic] = 100(, m(ARF = 60(
and [pic] = 3:1. Find
a. [pic]
b. [pic]
c. [pic]
d. m(ANG e. m(B
f. m(BUA g. m(NGS
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2. m(BCD = 2.
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3. m(BCD + m(BAD = 3.
4. m(BCD + m(BAD = 4. Factor out GCF
5. m(BCD + m(BAD = = 5.
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