2nd Term Lap 2015 16 Unit On Circles
2nd Term Lap 2015 16 Unit On Circles
2nd Term Lap 2015 16 Unit On Circles
A B
I I can illustrate secants, tangents, segments and sectors of a circle. I can derive inductively the relations among chords, arcs, etc.
S Definitions:
A circle is a set of all points in a plane having the same distance from a given point in the plane, called the center.
A radius is a segment whose endpoints are the center and a point on the circle. (Note: radii is the popularly known plural form of radius.
radiuses is also acceptable. Refer from the Merriam-Websters Dictionary.)
In a circle, all radii are equal.
A chord is a segment whose endpoints are on the circle.
A diameter of a circle is a chord that passes through the center of the circle. The length of a diameter, ,is twice the length of a radius,
. .
The circumference, , of a circle is the distance around it. The circumference of a circle can be obtained by multiplying the diameter by
(pi). . or .
The area of circle is the area (of the plane) bounded by the circle. It can be obtained by multiplying the with the square of the radius.
or
The pentagon is inscribed in the circle. The circle is inscribed in the pentagon.
The circle circumscribes about the pentagon. The pentagon circumscribes about the circle.
E Answer the following:
Exercises: EASY Set A #1-9, p. 249, Set B, #1, 3, 5, Set C #1, 3, 5, AVERAGE #1-15, p. 250
P
A 1. Can a circle be considered an arc by itself? Why, or why not?
2. Is a semicircle region a sector of the circle, or a segment of the circle?
21 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
I I can (a) derive inductively the relations between chords and radii; and, (b) solve problems on circles.
Note: Some authors do not use subtraction and division but use negative numbers and reciprocals.
22 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Hy A Congruency Theorem: If the hypotenuse and an acute angle of a right triangle are equal to the corresponding parts of another,
then the two right triangles are congruent.
Hy L Congruency Theorem: If the hypotenuse and a leg of one right triangle are equal to the corresponding parts of another right
triangle, then the two are congruent.
Pythagorean theorem:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of
the legs. That is, if and are the lengths of the legs, and is the length of the a c
hypotenuse, then .
b
Definition: If P is a point on such that P divides into two equal parts, then P
is said to bisect . P is called the midpoint of and .
Definition: The midpoint of a segment is said to bisect the segment. The midpoint of A P B
a segment AB, or any line, plane, ray, or segment which contains the midpoint and
does not contain , is called a bisector of .
Definition: If two angles have a common vertex and a common side between them,
they are called adjacent angles.
Definition: If the sum of two angles equals 180o, they are said to be supplementary.
Linear pair theorem: If two adjacent angles and form a straight angle, then they are supplementary.
B
Given:
and form a linear pair
PROOF
STATEMENTS REASONS
1. and form a linear pair 1. Given
2. 2. Angle Addition Postulate
3. is a straight angle 3. Def. of linear pair
4. = 180o 4. Def. of straight angle
5. 5. Substitution (Statements 2 and 4)
6. and are supplementary 6. Def. of supplementary angles
Equal Linear Pair Theorem: If two lines intersect forming a linear pair of equal angles, then the lines are perpendicular.
C
Given: AB intersects CD at O so that .
2 1
A O B
Prove: AB CD
D
PROOF
STATEMENTS REASONS
1. 1+ 2= 180o 1. Linear Pair Theorem
2. 1= 2 2. Given
3. 1+ 2=2 1=180 3. Substitution (Statements 1, 2); Addition
4. 1=90o 4. MPE
5. AB CD 5. Def. of perpendicular lines
23 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Radius-Chord theorem: If a radius of a circle is perpendicular to a chord, then the radius bisects the chord.
Given: OP at point Q.
Prove: MQ = QN
Converse of Radius-Chord theorem: If a radius of a circle bisects a chord that is not a diameter, then that radius is
perpendicular to the chord.
Prove:
PROOF
STATEMENTS REASONS
1. radius OP bisects the chord MN at Q 1. Given
2. MQ = QN 2. definition of bisector
3. Draw OM and ON 3. by construction
4. OM = ON 4. definition: radius of a circle
5. OQ = OQ 5. identity
6. 6. SSS
7. 7. CPCTC
8. OP MN 8. Equal Linear Pair Theorem
24 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Sn LEARNING COMPETENCY TOPIC
With a rating of at least 75%, in a given set of exercises, Chords-of-Congruent-Circles Theorem
14 1. prove theorems related to chords.
2. solve problems on circles. Reference : Global Mathematics, Banaag and Quan, pp. 244-246
LEARNING ACTIVITIES
G FYI
Why is the phrase perfect circle redundant?
Theoretically, a circle must be perfect, or else it is not a circle.
The phrase perfect circle opens the possibility of an imperfect circle which could not be a circle at all.
S Definition: The distance from a point to a line is the length of the perpendicular from the point to the line.
Right Angle Postulate: All right angles are equal.
Prove: WY@AC
PROOF
STATEMENTS REASONS
1. Circle X circle B 1. Given
2. WX is a radius of circle X. AB is a radius of circle B. 2. definition of radius (from the figure)
3. WX=AB 3. Definition of congruent circles (Sts. 1, 2)
4. distance XZ = distance BD 4. Given
5. XZ is perpendicular to WY 5. Definition of distance (St. 4)
6. BD is perpendicular to AC 6. Definition of distance (St. 4)
7. XZW is a right angle BDA is a right angle. 7. Definition of perpendicular (Sts. 5,6)
8. XZW and BDA are right triangles 8. Definition of right triangle (St. 7)
9. XZW BDA 9. HyL Congruence (Sts. 8, 3, 4)
10. 10. CPCTC (St. 9)
11. XZ bisects WY BD bisects AC 11. Radius-Chord theorem (St. 5, 6)
12. WZ= WY. AD= A C 12. definition of bisector (St. 11)
13. WY= A C 13. substitution (St. 10, 12)
14. WY@AC 14. MPE (St. 13)
A
1. In the figure, AB is a diameter of circle O. If AC = AD, prove that 1 = 2.
2. Prove that diameters of a circle are its longest chords. [Hint: recall the inequality theorems.]
25 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
I
I can (1) prove theorems related to central angles, (2) derive inductively the relations among chords, arcs, and central angles, and (3) find the
measure of the unknown parts using the postulates and theorems on central angles.
S
Definition: A central angle of a circle is an angle whose vertex
is the center of the circle and the sides are radii of the circle.
Definition: The degree measure of an arc of a circle is equal to the degree measure of the central angle intercepting the arc.
Definition: Congruent arcs. Two arcs are congruent if they belong to the same circle or to congruent circles and have the same
measure.
The sum of the angles about a point is 360o. Because of the correspondence between central angles and arcs, this implies that a whole
circle or an angle of one revolution is equal to 360 o; a semicircle measures 180o.
Central Angle-Intercepted Arc Postulate: The degree measure of a central angle is equal to the degree measure of its
intercepted arc.
,
Arc Addition Postulate: In a circle, if point X is on RT
then RT
R T
Minor Arcs-Chords Congruency Theorem: In a circle or congruent circles, if two minor arcs are congruent, then the chords
subtended by the arcs are congruent.
(A proof is presented on page 257 of the textbook.)
26 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Converse of Minor Arcs-Chords Congruency Theorem: In a circle or congruent circles, if two chords are congruent, then the
minor arcs subtending the chords are congruent.
Given:
T TM
Chord
T is subtended by the arc T
is subtended by the arc TM
Chord TM .
Prove: T
TM
PROOF
STATEMENTS REASONS
1. T TM 1. Given
Chord T is subtended by the arc T
Chord TM is subtended by the arc TM .
2. Draw , T and
M. 2. Two points determine a line.
3. ,
T and M are radii of the circle 3. Definition of a circle.
4.
M 4. Radii of the same circle are congruent.
5.
T T 5. identity/ reflexive property
6. T M T 6. SSS
7. T M T 7. CPCTC
8. T M T 8. definition of congruent angles
9. T T and
TM M T 9. definition of the degree measure of an arc
10. T
TM 10. Substitution (Statements 8 & 9)
11. T TM 11. definition of congruent arcs
27 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
S Central Angle-Intercepted Arc Postulate: The degree measure of a central angle is equal to the degree measure of its intercepted
arc.
, then RT
Arc Addition Postulate: In a circle, if point X is on RT R T
Minor Arcs-Chords Congruency Theorem: In a circle or congruent circles, if two minor arcs are congruent, then the chords
subtended by the arcs are congruent.
Converse of Minor Arcs-Chords Congruency Theorem: In a circle or congruent circles, if two chords are congruent, then the
minor arcs subtending the chords are congruent.
Take note of the following:
P Central Angle-Minor Arc Congruency Theorem. In a circle or congruent circles, if two central angles are congruent, then their
corresponding minor arcs are congruent.
Given: M
M
intercepts M
intercepts
Prove: M
PROOF
STATEMENTS REASONS
1. M 1. Given
2. M 2. Definition of congruent angles
3. M
intercepts M
3. Given
intercepts
4. M M
4. Central Angle-Intercepted Arc Postulate
Converse Central Angle-Minor Arc Congruency Theorem. In a circle or congruent circles, if two minor arcs are congruent,
then their corresponding central angles are congruent.
Given:
PT
O intercepts
POT intercepts PT
Prove: O POT
28 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
PROOF
STATEMENTS REASONS
1.
PT 1. Given
2.
PT 2. Definition of congruent arcs
3. O intercepts
3. Given
POT intercepts PT
4. O
4. Central Angle-Intercepted Arc Postulate
POT PT
5. O POT 5. Substitution (Statements 2 & 4)
6. O POT 6. Definition of congruent angles
Central Angle-Chord Congruency Theorem: In a circle or congruent circles, if two central angles are congruent, then their
corresponding chords are congruent.
(A proof is presented on p. 258 of the textbook.)
Converse Central Angle-Chord Congruency Theorem: In a circle or congruent circles, if two chords are congruent, then their
corresponding central angles are congruent.
(A proof is presented on p. 259 of the textbook.)
Since AOB and 1 form a linear pair, they are supplementary. By subtraction, 1 = 180o 75o=
105o. Similarly, 2 and are supplementary, so 3= 105o.
2. Q is the center of a circle. ABD is a straight line. If CBD=65o, BQA=130o and AX is a Figure:
diameter, find
a) BAQ c) ABQ e)
b) CBA d) CBQ f)
Answers:
a) 25o c) 25o e) 130o
b) 115o d) 90o f) 230o
SUMMATIVE TEST 3
29 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
I I can (1) prove theorems related to inscribed angles, (2) derive inductively the relations among arcs, central angle and inscribed angles, and,
(3) find the measure of the unknown parts using the theorems on inscribed angles.
S Definition: An inscribed angle is an angle whose vertex lies on the circle and the sides are chords of the circle.
Definition: An exterior angle of a triangle is an angle formed by one side of the triangle and an extension of another side.
In FIGURE 1 below, BCD is an exterior angle composed by side BC of , and side CD. CD is an extension of side AC.
Definition: An interior angle of a triangle is an angle determined by two sides of the triangle. In FIGURE 1 below, BAC, ABC, and
ACB are interior angles.
Definition: An interior angle of a triangle which is not adjacent to an exterior angle of the same triangle is called a remote interior angle
with respect to the exterior angle. In FIGURE 1 below, BAC and ABC are remote interior angles of exterior angle BCD.
FIGURE 1
In the figure,
Inscribed Angle Theorem: The measure of an inscribed angle is one-half the measure of its intercepted arc.
Case 1 Case 2 Case 3
Center on one side of the inscribed angle Center in the interior of the inscribed Center on the exterior of the inscribed
angle angle
30 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
PROOFS
Case 1
Given: Circle O with inscribed angle BAC: AC is a
diameter
Prove:
PROOF
STATEMENTS REASONS
1. Circle O with inscribed angle BAC; AC is a diameter. 1. Given
2. In ABO, . 2. Exterior Angle Theorem
3. OA = OB 3. Definition of radius of a circle
4. 4. Isosceles Triangle Theorem
5. 5. Substitution
6. 6. Addition
7. 7. MPE (multiplication with )
8. O 8. Substitution
9. O 9. Central Angle Intercepted Arc Postulate
Case 2
Given: Circle O with inscribed angle BAC; AD is a
diameter.
Prove:
PROOF
STATEMENTS REASONS
1. Circle O with inscribed angle BAC; AD is a diameter. 1. Given.
2. 2. Case 1
3. 3. Case 1
4. 4. APE (Statements 2 & 3)
5. ( ) 5. Factoring
6. ( ) 6. Substitution (Statements 4 & 5)
7. () 7. Arc-Addition Postulate
8. ( ) 8. Substitution (Statements 6 & 7)
9. 9. Angle-Addition Postulate
10. ( ) 10. Substitution (Statements 8 & 9)
31 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Case 3
Given: Circle O with inscribed angle BAC; AD is a
diameter.
Prove:
PROOF
STATEMENTS REASONS
1. Circle O with inscribed angle BAC; AD is a diameter. 1. Given.
2. 2. Case 1
3. 3. Case 1
4. 4. Angle Addition Postulate
5. 5. Substitution (Statements 2, 3 & 4)
6. ( ) 6. Factoring
Since the measure of the central angle of a circle is equal to the measure of its intercepted arc by the Central Angle-Intercepted
Arc Postulate, then the Inscribed Angle Theorem can be re-stated as follows: The measure of an inscribed angle is one-
half the measure of the central angle intercepting the same arc.
PROOF
STATEMENTS REASONS
1. In circle T, CDM is inscribed in the semicircle MNC. 1. Given
2. M
MN 2. Inscribed Angle Theorem
3.
MN 3. A semicircle measures 180o.
4. M ( ) 4. Substitution
5. M 5. multiplication
6. CDM is a right angle. 6. Definition of right angle.
32 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Example Exercises:
Answers:
33 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
I I can (1) prove theorems related to inscribed angles, (2) derive inductively the relations among arcs, central angle and inscribed angles, and,
(3) find the measure of the unknown parts using the theorems on inscribed angles.
Definition: An inscribed angle is an angle whose vertex lies on the circle and the sides are chords of the circle.
Inscribed Angle Theorem: The measure of an inscribed angle is one-half the measure of its intercepted arc.
S The measure of an inscribed angle is one-half the measure of the central angle intercepting the same arc.
Semicircle Theorem: An angle inscribed in a semicircle is a right angle.
Proof:
By the Inscribed Angle Theorem, and
By addition,
= ( )
Since the sum of is the whole circle, and one revolution is equal to 360 0, this implies that
( ) .
Similarly, Q S o.
34 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Since BD is a diameter,
= 1800 - 500 = 1300 and BA = 1800 - 1100 = 700.
By the Inscribed Angle Theorem, 3 35 0 and 4 65 0 .
The results imply that ADC 25 0 35 0 60 0 and by the
Cyclic Quadrilateral Theorem, CBA 180 0 60 0 120 0 .
BAD BCD 90 0 since both are subtended by
semicircles.
1.) O is the center of the circle. Find all the angles you can 2.) AB is a diameter of circle O. If 3.) In the figure below, AOB = 900,
under the following conditions: =800 and =1300, find the and BOC = 700. Find:
Given: ABF = 1000, BCG = 550, and CFG = 450. measures of angles 1 to 10.
y=
=
A
Perform the following:
35 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
I I can illustrate tangents to a circle. I can determine the relationship formed by tangent lines and other parts of the circle. I can prove theorems
on tangent lines and circles.
S
In the figure at the right , Q is the center of the circle. = 700. Find
and Q
Answer:
Prove: TV KR
One way to approach this problem is to use an indirect proof. In an indirect proof, an assumption is made which leads to a contradiction
and indirectly proves that the opposite is true.
36 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
ASSUMPTION:
PROOF
STATEMENTS REASONS
1. 1. Given assumption
2. T T 2. Right angles are equal.
3.
T
T 3. identity
4. TCV TCD 4. SAS
5.
T
T 5. CPCTC
Since
T is a radius of a circle and
T
T , then D must be on the circle. If D and V are both on the circle and on the tangent,
the tangent intersects the circle at two points. This contradicts the definition of a tangent and the original assumption that T
must be false.
Therefore, T
, and T R.
Converse of Radius-Tangent theorem: If a line is Figure:
perpendicular to a radius at its point of tangency on the
circle, then it is tangent to the circle.
Restatement: In circle T, if line RK is perpendicular to the
radius TV at V, then line RK is tangent to the circle.
Given: In circle T, line RK is perpendicular to the radius
TV at V.
Prove: RK is tangent to circle T.
Paragraph Proof:
Since RK TV, TVK is a right angle. Thus, in TVP, TVP > TPV and so TP > TV since it is opposite the greater angle. But, TV
is a radius. Therefore, P is outside the circle.
Similarly, any point S on line RK different from V is outside the circle. Therefore line RK has one and only one point in common with
the circle and it is the tangent.
NOTE: In a paragraph proof, the statements and reasons appear in sentences within a paragraph.
37 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Answers:
a) m APO=32.5o, m BPO=32.5o, and m AOB=115o b) m APB=60o c) m AOB=120o, m ABP=60o
a) If O
b) If is twice , find , , m O
T
Answers: a) O b) , ,m O o and
T o.
b) If and the
perimeter of ABC=52, find AB, AC, and BC.
38 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
I I can determine the relationship formed by tangent and secant lines, and I can prove theorem on tangents and secants.
S Definition: A tangent is a line that has exactly one point in common with a circle. The point of intersection is called the point of
tangency.
Definition: A secant is a line that intersects a circle in two distinct points. A secant determines a chord of a circle. A chord
determines a secant.
Radius-Tangent theorem: If a line is tangent to a circle, then it is perpendicular to the radius at its point of tangency.
Converse of Radius-Tangent theorem: If a line is perpendicular to a radius at its point of tangency on the circle, then it is tangent
to the circle.
Two Tangents External Point Theorem: If two tangent segments are drawn to a circle from an external point, then
(i) the tangent segments are congruent, and
(ii) the line joining the external point and the center of the circle bisects the angle formed by the two tangents.
P Take note of the following:
Tangent-Secant on the CircleIntercepted Arc Figure:
Theorem
The measure of angles formed by the intersection of a
tangent and a secant on the circle is half the measure of
the intercepted arc.
39 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
1. Find the unknown marked angles or arcs in each figure. 2. ABC is isosceles with AC=CB. A=70o.
a) b) DBE is a tangent at B. Find C, 1 and 2.
3.) If ABRS is a straight line through the center O; PQR is a straight 4.) ABC is tangent to the circle O at B; DB and EB are chords, BF is
line, RQB = 400 and SRQ = 1500. Find QBR, A, QBA, and a diameter. If = 1000 and ABD = 350, find EBC, E, BOD,
BPQ . , , and .
3.) ABCD is an inscribed quadrilateral. If A = (2x + 12)0 and C = 4.) ABCD is an inscribed quadrilateral. If A = (4x + 5)0, D = (7x +
(3x + 18)0, find A and C. If D = (2x 15)0, find B. 2)0 and C = (5x 2)0, find B.
40 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Example Sentence:
The students complained about the professor, because he always goes off on a tangent and starts talking about unrelated
things.
I I can (a) determine the relationship formed by tangent and secant lines, (b) prove theorem on tangents and secants, and (c) find the measure
of the unknown using theorems on tangents and secants.
S Radius-Tangent theorem: If a line is tangent to a circle, then it is perpendicular to the radius at its point of tangency.
Converse of Radius-Tangent theorem: If a line is perpendicular to a radius at its point of tangency on the circle, then it is tangent
to the circle.
Two Tangents External Point Theorem: If two tangent segments are drawn to a circle from an external point, then
(i) the tangent segments are congruent, and
(ii) the line joining the external point and the center of the circle bisects the angle formed by the two tangents.
Tangent-Secant on the CircleIntercepted Arc Theorem: The measure of angles formed by the intersection of a tangent and a
secant on the circle is half the measure of the arc.
Angles formed by two Tangents Theorem:
The measure of an angle formed by the intersection of two tangents on the exterior of the circle is equal to half the difference of the
measures of the major and minor arcs.
Angles Formed by Tangent-Secant-Exterior Point Theorem:
The measure of an angle formed by the intersection of a secant and a tangent on the exterior of the circle is half the difference of the
measures of their intercepted arcs.
PROOF
STATEMENTS REASONS
1. Draw chord XY. 1. by construction
2. FHY is an exterior angle of HXY 2. Definition of exterior angle of a triangle
3. m FHY = m HXY + m HYX 3. Exterior Angle Theorem
4. m HXY = () 4. Inscribed Angle Theorem
5. m HYX = () 5. Inscribed Angle Theorem
6. m FHY = () () 6. Substitution
7. m FHY = ( ( ) ( )) 7. Factoring
41 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
PROOF
STATEMENTS REASONS
1. and are secants intersecting in the 1. Given
exterior of circle O.
2. Draw segment KZ. 2. by construction
3. 1 is an exterior angle of KXZ. 3. Definition of an exterior angle of a triangle
4. 4. Exterior Angle Theorem
5. 5. APE
6. 6. Inscribed Angle Theorem
7. 7. Inscribed Angle Theorem
8. 8. Substitution
9. ( () ( )) 9. Factoring
Answers: Answers:
3.) TB is tangent to circle O at B. AOB is a diameter. EACT and EFB 4.) PQR and RST are secants to the circle. Chords PS and QT meet
are secants. If = 1100 and 1 = 250, find 2, 3, , at D. If R = 300 and PDT = 820, find , , 1, 2, and
E, ACB, ABC, EAB, and 4. PSR.
Answers: Answers:
42 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
A 5.) If x + y = 1200 and z = 200, find x and y. 6.) EDC and ABC are secants to the circle; BE and BD are chords. If
ECF = 1500, = 500, and BDC = 800, find all the marked angles
and arcs.
7.) ABC and EDC are secants meeting at C. If 2 = 250, C = 300, 3. PAB and RDC are straight lines. DB and AC are chords. Find all
find 1 and 3. the angles you can if PAD = 1080, DAC = 500, and DBA = 35.
SUMMATIVE TEST 4
43 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
I I can (a) determine the relationship formed by tangent and secant lines, (b) prove power theorems, and (c) find unknown segments using the
power theorems.
S AAA Similarity Theorem: Two triangles with corresponding angles equal are similar.
AA Similarity Theorem: Two triangles are similar if two angles of one are equal to the corresponding angles of the other.
Angles Formed by Tangent-Secant-Exterior Point Theorem:
The measure of an angle formed by the intersection of a secant and a tangent on the exterior of the circle is half the difference of the
measures of their intercepted arcs.
Angles Formed by Two Secants-Exterior Point Theorem:
The measure of an angle formed by two secants intersecting at the exterior of the circle is equal to half the difference of the major
arc and the minor arc.
Angles Formed by Two Secants-Interior Point Theorem:
The measure of an angle formed by two secants intersecting in the interior of the circle is equal to half the sum of the measures of
the arcs intercepted by the angles and the angle vertical to it.
P Take note of the following:
The relationship of a point to two intersecting lines and a circle is referred to as power, which was used by Jacob Steiner in 1826.
VM x VL = constant
VN x VQ = constant
Therefore, VM x VL = VN x VQ
Statement Reason
1. In circle C, VL and VQ are secants intersecting at an Given
external point V.
2. Determine LN to form VLN. Definition of a Triangle
Determine MQ to form VQM.
3. m MQN = ( ) Inscribed Angle Theorem
m NLM = ( )
4. m MQN m NLM Transitive Property of Equality
5. MQN NLM Definition of Congruent Angles
6. V = V Reflexive Property
7. VLN VQM AA Similarity
8. Corresponding Sides of Similar Triangles are in
Proportion
9. VM x VL = VN x VQ Definition of Proportionality
44 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Example: Solution:
AP x PB = CP x PD
6 x PB = 4 x 9
6PB = 36
PB = 6
Example: Solution:
45 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Example: Solution:
Step 2: (PB)2 = PC x PD
(PB)2 = 4 (16)
(PB)2 = 64
PB =
PB = 8
In the figure at the right, O is the center of the circle with diameter
. is tangent to the circle at B. If AB = 12, m C = 450, what is
CD?
46 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Does the word power in this quotation mean the same in the Power Theorems? How would you apply this quotation in real life when it comes
to gaining knowledge, holding a leadership position, and/or being esteemed by the society?
S 1.) PBA and PDC are secants to circle O. 2.) Chords AB and CD intersect at R. If AB = 3.) TA is tangent to circle O and TBC is a
If PB = 6 and PA = 24, What is the power 24, AR = 18, and CD = 21, find the power of secant. If RT = 9 and BC = 7, what is the
of P relative to O? R. power of T?
Figure: Figure: Figure:
If O and Q are the centers of two circles, the segment OQ is called the line of centers or segment of centers. The length of OQ in relation
to the radii of the circles will determine their relationship.
a.) b.) c.)
In the figure above, OQ r1 + r2; we say In the figure above, OQ r1 + r2; we say that In the figure above, OQ = 0, the circles
the two circles intersect each other. the two circles are apart, they have no point in have the same center and are said to be
common. concentric.
d.) e.)
In the figure above, OQ = r1 + r2; circle O touches circle Q at P. XY In the figure above, OQ = r1 r2.The circles touch at S and they are
is the tangent to both circles at P. The two circles are on opposite on the same sides of their tangent, ST. Circles O and Q are said to
sides of the tangent line. The circles are said to be tangent be tangent internally.
externally.
47 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Definition: Common tangent: A line or segment tangent to two distinct circles in a plane.
Definition: Common internal tangents: A common tangent to two circles in a plane that intersects their line of centers.
Definition: Common external tangents: A common tangent to two circles in a plane that does not intersect their line of centers.
Definition: Tangent Circles: Circles that touch each other at exactly one point.
Tangent Circles theorem: If two circles are tangent internally or externally, then their lines of centers pass through the point of
tangency.
Examples:
1.) Circles A and B have radii of (a.) internally (b.) externally
length 7 cm and 9 cm respectively.
How long is AB if the two circles are
tangent (a.) internally?
(b.) externally?
Solution: Solution:
Answer: Answer:
48 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
2.) Circles P, Q, and R are tangent externally to each other. If their 3.) In the figure below, circle O has a radius of 5 cm, circle Q has a
radii are 4 cm, 6 cm, and 3 cm respectively; find the perimeter of radius of 10 cm and OQ = 23 cm. PR is a common external tangent.
PQR. Calculate the length of PR.(Hint: construct rectangle POXR.)
Figure: Figure:
Solution/Answer: Solution/Answer:
Finding the length, L, of a common internal tangent of two circles given the distance, D, between the centers; and their radii, r1
and r2. (Derivation of the formula on page 281 of the textbook.)
Figure: is a common internal tangent of circles O and C.
O is the line of centers.
Let P P , O OP P
Since OA and CB are radii, they are perpendicular to the tangent at A and B, respectively.
OAP and CBP are right angles.
OAP= CBP, since right angles are equal.
APO= BPC, since they are vertical angles.
By AA similarity, PO P , thus, AOP= BCP.
Consider the following similar triangles from the figure above:
:
Hence,
( ) ( )
( )
( ) ( )
( )
49 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( ) [ ( ) ] ( ) [ ( ) ]
( ) ( )
( ) [ ( ) ] ( ) [ ( ) ]
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( )
A 1. In the figure, XY is a quadrant touching XZ at X and the quadrant 2. C is a point on AB such that AC=14 cm, CB=6 cm. Find the radius
TY at Y. If Z=90o, XZ=16 cm and ZT=4 cm, find the radii of the of a circle which touches AB at C and also touches the semicircle on
circles. AB as diameter.
Figure: Figure:
50 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
What makes a circular definition circular? Does the word circular (as it is used here) conform with the mathematical definition of a circle? Why,
or why not?
S Definition: Common tangent: A line or segment tangent to two distinct circles in a plane.
Definition: Line of centers: The segment joining the centers of two circles in a plane.
Definition: Common internal tangents: A common tangent to two circles in a plane that intersects their line of centers.
Definition: Common external tangents: A common tangent to two circles in a plane that does not intersect their line of centers.
Definition: Tangent Circles: Circles that touch each other at exactly one point.
Tangent Circles theorem: If two circles are tangent internally or externally, then their lines of centers pass through the point of
tangency.
P Take note of the following: (Reference: Holtmath 11, Holt, Rinehart and Winston of Canada, Limited. Toronto, 1988)
Three rolls of newsprint are bound together by a steel band as shown. The centers of two pulleys are 32 cm apart and each pulley has a
How long is the steel band if each roll has a 2 m diameter? diameter of 14 cm. What is the length of the belt around them?
Solution: Solution:
51 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
Name: ________________________________________ LG: __________ Date: ____________ATENEO DE NAGA UNIVERSITY HIGH SCHOOL
In two concentric circles, a tangent to the inner circle is a 12-cm chord A spherical watertank with a diameter of 8 m is held by supports as
of the outer circle. If the radius of the inner circle is 5 cm, what is the shown. The length of the water pipe from D to the ground B is 18 m.
radius of the outer circle? Find the lengths of and which are tangent to the tank.
Solution: Solution:
Solution: Solution:
Answer: Answer:
SUMMATIVE TEST 5
52 B. Beltran/W. Niebres/M. Aven/ LAP Grade 10 Mathematics SY 2015-16 ATENEO DE NAGA UNIVERSITY HIGH SCHOOL