modeling with geometry - unbounddanielsroar.weebly.com/.../ncm3_geometry_booklet.pdf · d a oid ses...

Modeling with

Geometry

6.3 Parallelograms

https://mathbitsnotebook.com/Geometry/Quadrilaterals/QDParallelograms.html

Properties of Parallelograms

Sid

es

A parallelogram is a quadrilateral with both pairs of opposite sides

parallel.

If a quadrilateral is a parallelogram,

the 2 pairs of opposite sides are congruent.

An

gle

s


the 2 pairs of opposite angles are congruent.


the consecutive angles are supplementary.

If a quadrilateral is a parallelogram and one angle is a right angle,

then all angles are right angles.

Dia

go

na

ls


the diagonals bisect each other.


the diagonals form two congruent triangles.

Example 1: Given: ▭ABCD is a parallelogram.

Prove: AB = CD and BC = DA.

Statement Reason

1. ABCD is a parallelogram 1.

2. 2. Definition of a parallelogram

3. <1 = <4, <3 = <2 3.

4. AC = AC 4.

5. ∆ABC = ∆CDA 5.

6. 6. CPCTC

1

Example 2: Given: ▭ABCD is a parallelogram.

Prove: AC and BD bisect each other at E.

Statement Reason

1. ABCD is a parallelogram 1. Given

2. AB || DC 2.

3. <1 = <4, <2 = <3 3.

4. AB = DC 4.

5. 5. ASA

6. AE = CE, BE = DE 6.

7. 7. Definition of bisector

Example 3: For what values of x and y must each figure be a parallelogram?

a) b)

c) d)

e) f)

2

Homework 6.3: Parallelograms Name: __________________________

Math 3

1. Use the diagram below to solve for x and y if the figure is a parallelogram.

a) PT = 2x, QT = y + 12,

TR = x + 2, TS = 7y

b) PT = y, TR = 4y -15,

QT = x + 6, TS = 4x - 6

2. Find the measure of each angle if the figure is a rhombus.

a) Find the m∠1. b) Find the m∠2.

c) Find the m∠3. d) Find the m∠4.

3. Solve for x if the figure is a rhombus. 4. Solve for x if the figure is a rectangle.

5. What is the length of LN if the figure is a

rectangle?

6. Solve for the missing angle measures if the figure

is a rhombus.

7. What is the length of SW?

8. Solve for x if the figure is a rhombus

.

3

6.4 Quadrilaterals

https://mathbitsnotebook.com/Geometry/Quadrilaterals/QDRectangle.html

Rectangle Rhombus Square A rectangle is a parallelogram with

four right angles.

A rhombus is a parallelogram with four

congruent sides.

A square is a parallelogram with four

congruent sides and four right angles.

A rectangle has all the properties of a

parallelogram PLUS:

4 right angles

Diagonals are congruent

A rhombus has all the properties of a

parallelogram PLUS:

4 congruent sides

Diagonals bisect angles

Diagonals are perpendicular

A square has all the properties of a

parallelogram PLUS:

All the properties of a rectangle

All the properties of a rhombus

Example 1: Solve for x and the measure of each angle if ▭DGFE is a rectangle.

Example 2: ▭ABCD is a rectangle whose diagonals intersect at point E.

a) If AE = 36 and CE = 2x – 4, find x.

b) If BE = 6y + 2 and CE = 4y + 6, find y.

Example 3: Using the diagram to the right to answer the following if ▭ABCD is a rhombus.

a) Find the m∠1. b) Find the m∠2.

c) Find the m∠3. d) Find the m∠4.

Example 4: Solve for each variable if the following are rhombi.

a) b)

4

Tra

pe

zoid

A t

rap

ezo

id is

a q

ua

drila

tera

l with

exa

ctly o

ne

pa

ir o

f p

ara

llel s

ide

s, c

alle

d b

ase

s, a

nd

tw

o

no

np

ara

llel s

ide

s, c

alle

d le

gs.

Isosceles Trapezoids Trapezoid Midsegment

An isosceles trapezoid is a trapezoid with

congruent legs.

The median (also called the midsegment) of a

trapezoid is a segment that connects the

midpoint of one leg to the midpoint of the other

leg.

A trapezoid is isosceles if there is only:

One set of parallel sides

Base angles are congruent

Legs are congruent

Diagonals are congruent

Opposite angles are supplementary

Theorem: If a quadrilateral is a trapezoid, then a)

the midsegment is parallel to the bases and b)

the length of the midsegment is half the sum of

the lengths of the bases

Example 5: CDEP is an isosceles

trapezoid and m<C = 65. What

are m<D, m<E, and m<F?

Example 6: What are the values

of x and y in the isosceles triangle

below if DE || DC?

Example 7: QR is the midsegment of

trapezoid LMNP. What is x and the

length of LM?

You Try! TU is the midsegment

of trapezoid WXYZ. What is x

and the length of TU?

Kit

e

A k

ite

is

a q

ua

drila

tera

l with

two

pa

irs

of

ad

jac

en

t,

co

ng

rue

nt

sid

es.

If a quadrilateral is a kite, then:

Its diagonals are

perpendicular.

Its diagonals bisect the

opposite angles.

One pair of opposite

angles are congruent.

One diagonal bisects the

other.

Example 4: Quadrilateral DEFG is a kite. What are m<1,

m<2, and m<3?

You Try! Quadrilateral KLMN is a kite. What are m<1,

m<2, and m<3?

5

Homework 6.4: Quadrilaterals

Directions: For questions #1-2, find the measure of each missing angle.

1. 2.

Directions: For questions #3-4, find x and the length of EF.

3. 4.

Directions: For questions #5-6, find the measures of the numbered angles in each kite.

5. 6.

Challenge Question: Solve for the unknown angle measures in the kite shown below.

6

Math 3 6.7 Tangent Lines of Circles Unit 6

SWBAT solve for unknown variables using theorems about tangent lines of circles.

Tangent to a Circle Ex: (AB)

A line in the plane of the circle that intersects the

circle in exactly one point.

Ex: Segment AB is a tangent to Circle O.

Point of Tangency The point where a circle and a tangent intersect.

Ex: Point P is a point of tangency on Circle O.

Tangent Theorem 1: Converse Theorem 1:

If a line is tangent to a circle, then it is perpendicular

to the radius draw to the point of tangency.

If a line is perpendicular to the radius of a circle at its

endpoint on a circle, then the line is tangent to the

circle.

Example: If RS is tangent, then PR _____ RS.

Example 1: Find the measure of x.

a) b)

Example 2: Find x. All segments that appear tangent are tangent to Circle O.

a) b)

Example 3: Is segment MN tangent to Circle O at P? Explain.

7

Tangent Theorem 2: If two tangent segments to a circle share a common endpoint outside the circle,

then the two segments are congruent.

Example 4: Solve for x.

Circumscribed vs. Inscribed

To circumscribe is when you draw a figure

around another, touching it at points as

possible.

To inscribe is to draw a figure within

another so that the inner figure lies

entirely within the boundary of the

outer.

Ex: The circle is circumscribed about the

triangle. Ex: The triangle is inscribed in the circle.

Tangent Theorem 3:

(Circumscribed Polygons)

When a polygon is circumscribed about a circle, all of the sides of the polygon

are tangent to the circle.

Example 5: Triangle ABC is circumscribed about ⊙O. Find the perimeter of triangle ABC.

You Try! Find x. Assume that segments that appear to be tangent are tangent.

a) b b) c)

8

Practice 6.7: Tangents of Circles

Directions: Assume that lines that appear to be tangent are tangent. O is the center of each circle. What is the

value of x?

1. 2. 3.

Directions: In each circle, what is the value of x to the nearest tenth?

4. 5. 6.

7. TY and ZW are diameters of S. TU and UX are tangents of S. What is mSYZ?

Directions: Each polygon circumscribes a circle. What is the perimeter of each polygon?

8. 9.

10. 11.

9

6.8 Chords & Arcs of Circles

https://mathbitsnotebook.com/Geometry/Circles/CRChords.html

Example 1: Name the circle, a radius, a chord, and a diameter of the circle.

Circle: ___________ Circle: ___________

Radius: ___________ Radius: ___________

Chord: ____________ Chord: ____________

Diameter: __________ Diameter: __________

Since a _________________ is composed of two radii, then d = 2r and r = d/2

Theorem 1: Converse Theorem 1:

Within a circle or in congruent circles, chords

equidistant from the center or centers are

congruent.

Within a circle or in congruent circles, congruent

chords are equidistant from the center (or

centers).



central angles have congruent arcs.


arcs have congruent central angles.



central angles have congruent chords.


chords have congruent central angles.



chords have congruent arcs.


arcs have congruent chords.

Example 2: The following chords are equidistant from the center of the circle.

a) What is the length of RS? b) Solve for x.

A _____________ that passes through

the center is a ______________ of a

circle.

Any segment with

_________________ that are the

center and a point on the circle is a

_______________. Any segment with

____________________ that are

on a circle is called a

_______________.

The given point is called the

_____________.

This point names the circle.

10

Theorem 5:

In a circle, if a diameter is perpendicular to a

chord, then it bisects the chord and its arc.

Theorem 6:

In a circle, if a diameter bisects a chord that is not

a diameter, then it is perpendicular to the chord.

Theorem 7:

In a circle, the perpendicular bisector of a chord

contains the center of the circle.

Example 3: In ⊙O, 𝐶𝐷̅̅ ̅̅ ⊥ 𝑂𝐸̅̅ ̅̅ , 𝑂𝐷 = 15, 𝑎𝑛𝑑 𝐶𝐷 = 24. Find x.

Example 4: Find the value of x to the nearest tenth.

You Try! Find the value of x to the nearest tenth.

a) b)

11

Practice 6.8: Chords & Arcs of Circles

6. A student draws X with a diameter of 12 cm. Inside the circle she inscribes equilateral ∆ABC so that

AB , BC , and CA are all chords of the circle. The diameter of X bisects AB . The section of the

diameter from the center of the circle to where it bisects AB is 3 cm. To the nearest whole number,

what is the perimeter of the equilateral triangle inscribed in X?

12

6.9 Inscribed Angles

https://mathbitsnotebook.com/Geometry/Circles/CRAngles.html

Major Arc: Minor Arc: Semicircle:

An arc of a circle

measuring more than or

equal to 180˚

An arc of a circle

measuring less than 180˚

An arc of a circle

measuring 180 ˚

Central Angle: A central angle is an angle formed by two intersecting radii such that

its vertex is at the center of the circle.

Central Angle

Theorem:

In a circle, or congruent circles, congruent central angles have

congruent arcs.

Example 1: Identify the following in ☉P at the right. For parts d-f, find the measure of each arc in ☉P.

a) A semicircle b) A minor arc c) A major arc

d) 𝑆�̂� e) 𝑆𝑇�̂� f) 𝑅�̂�

Inscribed Angle: An inscribed angle is an angle with its vertex "on" the circle, formed

by two intersecting chords.

Inscribed Angle

Theorem:

The measure of an inscribed angle is half the measure of its

intercepted arc.

Example 2: What are the values of a and b? You Try! What are the m∠𝐴, m∠𝐵, m∠𝐶, and m∠𝐷?

13

Corollary 1: Corollary 2: Corollary 3:

Two inscribed angles that intercept

the same arc are congruent.

An angle inscribed in a semicircle is

a right angle.

The opposite angles of a

quadrilateral inscribed in a circle are

supplementary.

Example 3: What is the measure of each numbered angle?

a) b)

You Try! Find the measure of each numbered angle in the diagram to the right.

a) 𝑚∡1 = b) 𝑚∡2 =

c) 𝑚∡3 = d) 𝑚∡4 =

Tangent Chord

Angle:

An angle formed by an intersecting tangent and chord has its vertex

"on" the circle.

Tangent Chord

Angle Theorem:

The tangent chord angle is half the measure of the intercepted arc.

Tangent Chord Angle = ½ (Intercepted Arc)

Example 4: In the diagram,

𝑆𝑅 ⃡ is tangent to the circle at

Q. If 𝑚𝑃𝑀�̂� = 212, what is

the 𝑚∠𝑃𝑄𝑅?

You Try! In the diagram, 𝐾𝐽 ⃡ is tangent to ⊙ 𝑂. What are

the values of x and y?

Practice: Find the value of each variable. For each circle, the dot represents the center.

1. 2. 3.

4. 5. 6.

14

Homework 6.9: Inscribed Angles Name: __________________________

Math 3

Directions: Find the value of each variable. For each circle, the dot represents the center.

1. 2. 3.

4. 5. 6.

Directions: Find the value of each variable. Lines that appear to be tangent are tangent.

7. 8. 9.

Directions: Find each indicated measure for M.

10. mB 11. mC

12. 13.

Directions: Find the value of each variable. For each circle, the dot represents the center.

14. 15. 16.

15

6.9 Angle Measures and Segment Lengths

https://mathbitsnotebook.com/Geometry/Circles/CRSegmentRules.html.

Theorem 1: Theorem 2:

The measure of an angle formed by two lines that

intersect inside a circle is half the sum of the

measures of the intercepted arcs.

The measure of an angle formed by two lines that

intersect outside a circle is half the difference of the

measures of the intercepted arcs.

Example 1: Find each measure.

a) b) c)

Example 2: Find the following angles.

a) m∠MPN b) c)

You Try! Find the following angles.

a) b) x c) Arc AB

16

Theorem 3:

For a given point and circle, the product of the lengths of the two segments from the point to the circle is

constant along any line through the point and the circle.

Example 4: Find the value of the variable in ⊙ 𝑂.

a) b) c)

You Try! What is the value of the variable to the nearest tenth?

17

Homework 6.9: Angles and Segments

Directions: Solve for x.

1. 2. 3.

Directions: Solve for each variable listed.

4. 5. 6.

7. There is a circular cabinet in the dining room. Looking in from

another room at point A, you estimate that you can see an

arc of the cabinet of about 100°. What is the measure of A

formed by the tangents to the cabinet?

Directions: Find the diameter of O. A line that appears to be tangent is tangent. If your answer is not a whole

number, round to the nearest tenth.

8. 9. 10.

Directions: CAand CB are tangents to O. Write an expression for each arc or angle in terms of the given

variable.

11. using x 12. using y 13. mC using x

18

6.10 Equations of Circles

https://mathbitsnotebook.com/Geometry/Equations/EQCircles.html

Example 1: Write the equation of a circle with the given information.

a) Center (0,0), Radius=10

h = k = r =

b) Center (2, 3), Diameter=12

h = k = r =

Example 2: Determine the center and radius of a circle the given equation.

a) 4

922 yx b) 81)5()3( 22 yx

c) 1)6()4( 22 yx

Example 3: Use the center and the radius to graph each circle.

a) 64)2( 22 yx b) 36)4( 22 yx

Center:

Radius:

Center:

Radius:

Example 4: Write the equation of a circle with a given center

(2, 5) that passes through the point (5 ,-1).

Writing an Equation with a

Pass-Thru Point

Step 1: Substitute the center (h, k) into the equation

Step 2: Substitute the “pass through point (x, y)”

into the equation for x and y.

Step 3: Simplify and solve for r2.

Step 4: Substitute r2 back into the equation

from Step 1.

Standard Form of Circles

Center:

Radius:

Point on the

circle:

19

Example 5: Write the equation of a circle with endpoints of diameter at (-6, 5) and (4, -3).

Writing the Equation of a Circle in Standard Form

Step 1: Group x’s and group y’s together.

Step 2: Move any constants to the right side of the equation.

Step 3: Use complete the square to make a perfect square trinomial for the x’s and then again for the y’s.

*Remember, whatever you do to one side of the equation, you must do to the other!

Step 4: Simplify factors into standard form of a circle!

Example 5: Write the equation of a circle in standard form. Then, state the center and the radius.

a) x2 + y2 + 4x - 8y + 16 = 0 b) x2 + y2 + 6x - 4y = 0

c) x2 + y2 - 6x - 2y + 4 = 0 d) x2 + y2 + 8x - 10y - 4 = 0

Writing an Equation with Two Points

on the Circle Midpoint Formula

Find the midpoint (radius) between the two endpoints, and

then follow steps 1-4.

20

Homework 6.10: Equations of Circles

Note: If r2 is not a perfect square then leave r in simplified radical form but use the decimal equivalent for

graphing. Example: 46.33212

1) Graph the following circle:

a. (x - 3)2 + (y + 1)2 = 4 b. (x – 2)2 + (y – 5)2 = 9 c. (y + 4)2 + (x + 2)2 = 16

2) For each circle, identify its center and radius.

a. (x + 3)2 + (y – 1)2 = 4 b. b. x2 + (y – 3)2 = 18 c. (y + 8)2 + (x + 2)2 = 72

Center:_____________

Radius:_____________

Center:___________

Radius:____________

Center:_____________

Radius:_____________

3) Write the equation of the following circles:

4) Give the equation of the circle that is tangent to the y-axis and center is (-3, 2).

5) Compare and contrast the following pairs of circles

a. Circle #1: (x - 3)2+ (y +1)2 = 25

Circle #2: (x + 1)2 + (y - 2)2 = 25

b. Circle #1: (y + 4)2+ (x + 7)2 = 6

Circle #2: (x + 7)2 + (y + 4)2 = 36

21

6) Find the standard form, center, and radius of the following circles:

a. x2 + y2 – 4x + 8y – 5 = 0 b. 4x2 + 4y2 + 36y + 5 = 0

Center:________ Radius:_________ Center:________ Radius:_________

7) Graph the following circles.

a. x2 – 2x + y2 + 8y – 8 = 0 b. x2 + y2 – 6x + 4y – 3 = 0

8) Give the equation of the circle whose center is

(5,-3) and goes through (2,5)

9) Give the equation whose endpoints of a diameter

at (-4,1) and (4, -5)

10) Give the equation of the circle whose center is

(4,-3) and goes through (1,5)

11) Give the equation whose endpoints of a diameter

at (-3,2) and (1, -5)

22

Length of a Circular Arc

Arcs have two properties. They have a measurable curvature based upon the corresponding central angle (measure of arc = measure of central angle). Arcs also have a length as a portion of the circumference.

portion of circle

whole circle

central angle in deg rees

360

central angle in radians

2

arc length

circumference

-or - For a central angle θ in radians, and arc length s - the proportion can be simplified to a

formula:

Examples: 1) For a central angle of π/6 in a circle of radius 10 cm, find the length of the intercepted

arc. 2) For a central angle of 4π/7 in a circle of radius 8 in, find the length of the intercepted

arc. 3) For a central angle of 40° in a circle of radius 6 cm, find the length of the intercepted

arc. 4.) Find the degree measure to the nearest tenth of the central angle in a circle that has an

arc length of 87 and a radius of 16 cm.

Remember:

circumference of a circle = 2πr

x

360

length CB

2r

x (radians)

2

length CB

2r

2

s

2r

s2 2r

s r

Length of an Arc: s = rθ

for θ in radians

23

Area of a Sector

Sector of a circle: a region bounded by a central angle and the intercepted arc

Sectors have an area as a portion of the total area of the circle.

portion of circle

whole circle

central angle in deg rees

360

central angle in radians

2

area of sec tor

area of circle

-or -

For a central angle θ in radians, and area of sector A, the proportion can be simplified to a

formula:

Examples:

5) Find the area of the sector of the circle that has a central angle measure of π/6 and a

radius of 14 cm.

6) Find the area of the sector of the circle that has a central angle measure of 60° and a

radius of 9 in.

HONORS

7) A sector has arc length 12 cm and a central angle measuring 1.25 radians Find the

radius of the circle and the area of the sector.

Remember:

area of a circle = πr2

x

360

area of sec tor

r2

x (radians)

2

area of sec tor

r2

2

A

r2

A2 r2

A 1

2r2

Area of a Circular Sector: A=½r2θ

for θ in radians

24

Practice: Arc Length & Area of Sectors

25

Review:

1. Find x

2.

3.

4.

5.

6. What is x?

7.

8. Find x.

9. Find x

10. Find HG

11. Find measure of arc x

12. Find CE

27

13. Which is the equation of a circle with r=11 and center (0,6)?

14. Find the arc length

15. Find x

16. Find the area of the sector

17. Find the center and radius

18. Find x

19. Find x and y

20. What is the length of RS?

21. Find WS

22. What is the measure of angle 1?

23. Find CA

24. Find x

28

modeling with geometry - unbounddanielsroar.weebly.com/.../ncm3_geometry_booklet.pdf · d a oid ses...

Documents