Geometry and Trig

GeometryGoals• Define a ray as originating at an endpoint, A, along a half-line containing a point B, as

• Represent an angle, , between two rays, with a common endpoint, A, by the counterclockwise rotation of ray onto ray

Big Idea

A line segment is a part of a line. A line is denoted as

A ray is a half-line and originates at a point.

Intersecting rays form an angle and the size of the angle is the measure of the angle and denoted

∠BAC ABu ruu

ABs ruu

m∠BAC

ABu ruu

ACu ruu

Geometry

m∠COB< 90° acute

m∠COB> 90° obtuse

m∠COB=90° right

m∠COB=180° straight

Goals• Define acute, obtuse, right, and straight angles.

Geometry

m∠AOB+m∠BOC =90° complementary

m∠EOF +m∠FOG=180° supplementary

For adjacent angles

Goals• Represent the complement of an angle with a geometric figure and calculate the complement.

• Represent the supplement of an angle with a geometric figure and calculate the supplement.

Geometry

Parallel lines

Intersecting lines

Goals• Prove and apply the fact that the congruence of vertical angles formed by intersecting lines.

These vertical angles are congruent.

Why?

Geometry

Intersecting lines

Goals• Prove and apply the fact that the congruence of vertical angles formed by intersecting lines.

These vertical angles are congruent.

m∠AOD+m∠DOB=180°m∠DOB+m∠BOC =180°∴m∠AOD=m∠BOC

Geometrytransversal

interior ∠AOC∠BOCexterior ∠AOD∠BOD

We won’t do an informal proof:

When parallel lines are cut by a transversal the alternate interior angles have equal measure.

alternate interior ∠AOC and ∠CO'F

Geometry

The measure of this angle is equal toWhy?

m∠BOD

Goals• Prove and apply the fact that the interior angles on opposite sides of a transversal of parallel lines are congruent.

• Prove and apply the fact that the corresponding angles formed by the transversal of parallel lines are congruent.

Geometry

HW due Tuesday 1/4250.3, 5, 6, 7, 9, 255.4, 6, 16, 27, 30, 38 and 262.23

Find the measures of all other angles.

ABs ruu

CDs ruu

and m∠5=40 and m∠4=30

Goals• Prove and apply the fact that the interior angles on the same side of a transversal of parallel lines are supplements.

GeometryGoals• Classify triangles as acute, equiangular, right, and obtuse.

GeometryGoals• Differentiate the legs and hypotenuse of a right triangle.

GeometryGoals• Classify triangles as scalene, isosceles, and equilateral.

GeometryGoals• Calculate one interior angle of a triangle if two are known.

a =A alternate interior angles

b=B alternate interior angles

a+C +b=180°∴A+C + B=180°

Geometry

First construction: Congruent sides

• Draw two rays that intersect• Use the compass to locate equal distances from S

GeometrySecond construction: Congruent angles

• Draw intersecting rays with endpoint S• Draw a ray with endpoint M• Use the compass to locate equal distances from S, A and B• Adjust the compass with point at B to intersect at A• With compass at L locate the point N

GeometryThird construction: Perpendicular bisector

• Draw intersecting ray containing points A and B• Use the compass to construct a circle centered on A• Construct a circle with the same diameter centered on B• Connect the intersections of the circles

GeometryThird construction: Angle bisector

• Draw rays containing points A and B intersecting at O• Use the compass to construct a circle centered on A• Construct a circle with the same diameter centered on B• Connect the intersections of the circles

GeometryGoals• Calculate the interior angle of a quadrilateral if three interior angles are known.

a +b+ c=180°x+ y+ z=180°a+b+ c+ x+ y+ z=360°Q=b+ y and P=c+ z∴a+Q+ x+ P =360°

GeometryGoals• Calculate the interior angle of an n-sided polygon if n-1 interior angles are known.

Within an n-sided polygon (n-2) triangles can be constructed and the sum of the angles in each is 180 so

The sum of the angles in an n-sided polygon is (n-2)180°

GeometryGoals• Apply mathematical reasoning to construct the area of an irregular polygon by decomposition into simpler geometric figures where possible.

A triangle can be inscribed in a rectangle. In this case there are two.Each has an area that is half of a rectangular area.

Area =

12

b−c( )h+12

c( )h=12

bh

GeometryGoals• Apply mathematical reasoning to construct the area of an irregular polygon by decomposition into simpler geometric figures where possible.

A piece of land is bounded by two parallel roads and two roads that are not parallel form a trapezoid. Along the parallel roads the land measures 1.3 miles and 1.7 miles. The distance between the parallel roads, measured perpendicular to the roads, is 282 miles.

a. Find the area of the land. Express the area to the nearest tenth of a mile.

b. An acre is a unit of area often used to measure land. There are 640 acres in a square mole. Express to the nearest hundred acres, the area of land.

GeometryA piece of land is bounded by two parallel roads and two

roads that are not parallel form a trapezoid. Along the parallel roads the land measures 1.3 miles and 1.7 miles. The distance between the parallel roads, measured perpendicular to the roads, is 282 miles.

a. Find the area of the land.

First draw a picture.

Geometry

Then work out a symbolic representation of the area.

ed =12

ae+12

ce+Area of trapezoid

a+ c=d-b

∴Area of trapezoid=ed-12

e(d−b) =12

e(d+b)

a. Find the area of the land.

First draw a picture.

Finally, evaluate the representation.

GeometryGoals• Apply mathematical reasoning to construct the area of an irregular polygon by decomposition into simpler geometric figures where possible.

ABCD is a quadrilateral. The diagonals are perpendicular at E.

Find the area of ABCD.

GeometryGoals• Calculate the surface area and volumes of rectangular and cylindrical solids, and spheres.

Object Area Volume

cube 6s2 s3

rect. solid 2WH+2WL+2LH LWH

cylinder 2πr2 +πrh πr2h

sphere 4πr2 4

3πr3

Some important areas and volumes are memorized formulas.

GeometryGoals• Calculate the surface area and volumes of rectangular and cylindrical solids, and spheres.

Some areas and volumes are constructed from these formulas.

What is the area of the region that isshaded black?

First you find the area of the regularhexagon and then you subtract it fromthe area of the circle.

GeometryGoals• Calculate the surface area and volumes of rectangular and cylindrical solids, and spheres.

Some areas and volumes are constructed from these formulas.

What is the volume of the prism?

Geometry

HW due Wednesday 1/5268.13, 14, 27, 31, 277.12, 278.19, 281.3, 4, 5, 8, 286.13, 291.13, 17, 292.19

Goals• Calculate the relative error in the measurement of a length, surface area or volume.

Percent of error =

observed amount - true amount

true amount×100

GeometryHomework Quiz #1

x + (2x+15) =90°3x=75°∴x=25°

Homework problem: The complement of an angle is 14 times as large as the angle. Find the complement.

Regents Exam question:

GeometryHomework Quiz #2

Homework problem: Do an informal proof that the interior angles when parallel lines are cut by a transversal are equal.

Regents Exam question:

x +5°=(2x−5)°x=10°

GeometryHomework Quiz #3

m∠5=40°∴m∠7 =40°m∠4=30°∴m∠2=30°∠6complements∠7∴m∠6=140°∠3complements∠2∴m∠3=150°∠9 alt.interior of ∠5∴m∠9=40°∠11 alt.interior of ∠4∴m∠11=30°m∠9+m∠10+m∠11=180°∴m∠10 =110°

ABs ruu

EFs ruu

and m∠5=40 and m∠4=30

TrigonometryGoals• Prove and apply the Pythagorean theorem.• Calculate the length of a hypotenuse from the lengths of the legs.

Area =(a+b)2

area=c2 =(a+b)2 −4•12

ab

=a2 + 2ab+b2 −2ab ∴c2 =a2 +b2

If the lengths of two sides of a right triangle are known thenthe length of the third side can be determined.

TrigonometryGoals• Prove and apply the Pythagorean theorem.• Calculate the length of a hypotenuse from the lengths of the legs.

c2 =a2 +b2

c =± c2

The negative root is rejected since the length is positive.

TrigonometryGoals• Calculate the length of a hypotenuse from the lengths of the legs.

c2 =a2 +b2

The negative root is rejected since the length is positive.

If the hypotenuse = 25 and one leg = 20, what Is the length of the other leg?

252 −202 =625−400 =225 225=25•9but

c =± 25•9 =± 52 •32 =±5•3=15

so

TrigonometryGoals• Represent the tangent of the acute angle of a right triangle as the ratio of the lengths of the opposite and adjacent sides.

TrigonometryGoals• Represent the tangent of the acute angle of a right triangle as the ratio of the lengths of the opposite and adjacent sides.

The leg OA is adjacent to AThe leg OB is adjacent to B

TrigonometryGoals• Represent the tangent of the acute angle of a right triangle as the ratio of the lengths of the opposite and adjacent sides.

The ratio of the adjacentand opposite sides remains constant for these similartriangles.

bB

bA=

cCcA

=dDdA

The tangent of the angle a is the ratio of the length of the opposite side to the length of the adjacent side.

TrigonometryGoals• Represent the tangent of the acute angle of a right triangle as the ratio of the lengths of the opposite and adjacent sides.

The tangent of the angle a, tan(a), is the ratio of the length of the opposite side to the length of the adjacent side.

tan(a) =

31

tan(a) =32

tan(a) =33

Trigonometry

HW due Thursday 1/6305.1, 2, 9, 11, 20, 28, 31, 34, 36, 311.3, 6, 13, 16, 17, 22, 312.30, 31, 35

Goals• Calculate the tangent of the acute angle of a right triangle.

311.3 In a right triangle with sides of length 3 and 3 find the tangents of both acute angles.

311.13 Calculate the tangent of 1°

311.16 Calculate the tangent of 67°

312.30 Calculate tan(20°) and tan(40°) to the nearest ten-thousandth. Does tan(a) double when the angle doubles?

TrigonometryGoals•Calculate the ratio of the opposite and adjacent sides of a right triangle using the inverse tangent of the acute angle.

The inverse tangent tan-1(a), is the answer to the question: if the ratio is known what is the angle?

71.56 =tan−1 3

1⎛

⎝⎜⎞

⎠⎟ 56.30 =tan−1 3

2⎛

⎝⎜⎞

⎠⎟ 45=tan−1 3

3⎛

⎝⎜⎞

⎠⎟

Trigonometry

HW due Thursday 1/6305.1, 2, 9, 11, 20, 28, 31, 34, 36, 311.3, 6, 13, 16, 17, 22, 312.30, 31, 35

Goals• Calculate the ratio of the opposite and adjacent sides of a right triangle using the inverse tangent of the acute angle.

311.17 Calculate the measure of the angle A if tan(A)=0.0875

311.22 Calculate the measure of the angle A if tan(A)=3.0777

312.31 in a right triangle with legs of length 6 and 6: a. find tangent of the acute angles. b. find the acute angle

TrigonometryHW Quiz #2

tan(C) =

oppositeadjacent

=815

TrigonometryHW Quiz #2

shaded area = rectangular area - triangular area

=MH•HT-1

2GB•HT =6•15−

124.6 •15=90−34.5=55.5

TrigonometryHW Quiz #2

3. AB and CD intersect at point E. m∠AEC=6x+20 and m∠DEB=10x.Wha t is t he value of ?x

[ ]A 5 [B] 10 [C] 21.25 [ ]D 4.375

A

B

C

D10x

6x+20 6x+20=10x4x=20x=5

E

TrigonometryApplications of the tangent

• if the opposite side and an angle are known the adjacent side can be determined

A stake is to be driven into the ground away from the base of a 50-foot pole. A wire from the stake to the top of pole is to make a 52° angle with the ground. Where is the stake?

TrigonometryApplications of the tangent

• if the opposite side and an angle are known the adjacent side can be determined

A stake is to be driven into the ground away from the base of a 50-foot pole. A wire from the stake to the top of pole is to make a 52° angle with the ground. Where is the stake?

Let x be the horizontal distance

tan(52°) =50 feet

x

x=50 feettan(52°)

=50 feet1.299

=39.06 feet

TrigonometryApplications of the tangent

• if the adjacent side and the angle of elevation is known then the length of the opposite side can be determined.

A tree casts a 25-foot shadow on a sunny day. If the angle formed by the tip of the tree and the tip of the shadow is 32° what is the height of the tree?

TrigonometryApplications of the tangent

• if the adjacent side and the angle of elevation is known then the length of the opposite side can be determined.

A tree casts a 25-foot shadow on a sunny day. If the angle formed by the tip of the tree and the tip of the shadow is 32° what is the height of the tree?

Let x be the vertical distance

tan(32°) =x

25 feetx=tan(32°) • 25 feet=0.625• 25 feet=15.6 feet

TrigonometryApplications of the tangent

• if the adjacent side and the angle of elevation is known then the length of the opposite side can be determined.

A tree casts a 25-foot shadow on a sunny day. If the angle formed by the tip of the tree and the tip of the shadow is 32° what is the height of the tree?

Let x be the vertical distance

tan(32°) =x

25 feetx=tan(32°) • 25 feet=0.625• 25 feet=15.6 feet

Trigonometry

HW due Thursday 1/10316.2, 3, 5, 9, 12, 13, 317.19, 21, 320.1, 3, 4, 5, 6, 8, 14, 20, 31, 32, 39, 322.43, 47, 50

In some problems you’re interested in making measurements of depth rather height. Then the “angle of elevation” is replaced with an “angle of depression.”

TrigonometryGoals• Represent the sine of the acute angle of a right triangle as the ratio of the lengths of the opposite side and the hypotenuse.

• Calculate the sine of the acute angle of a right triangle.

bB

AB=

cCAC

=dDAD

A second constant ratio

is called the sine of a,sin(a).

TrigonometryGoals• Represent the sine of the acute angle of a right triangle as the ratio of the lengths of the opposite side and the hypotenuse.

The sine of the angle a, sin(a), is the ratio of the length of the opposite side to the length of the hypotenuse.

sin(a) =

3

10 sin(a) =

3

13 sin(a) =

3

18

TrigonometryGoals•Calculate the ratio of the opposite side and the hypotenuse of a right triangle using the inverse sine of the acute angle.

The inverse sine sin-1(a), is the answer to the question: if the ratio is known what is the angle?

71.56 =sin−1 3

10

⎛

⎝⎜⎞

⎠⎟ 56.30 =sin−1 3

13

⎛

⎝⎜⎞

⎠⎟ 45=sin−1 3

18

⎛

⎝⎜⎞

⎠⎟

TrigonometryGoals• Represent the cosine of the acute angle of a right triangle as the ratio of the lengths of the adjacent side and the hypotenuse.

• Calculate the cosine of the acute angle of a right triangle.

Ab

AB=

AcAC

=AdAD

A third constant ratio

is called the sine of a,cos(a).

TrigonometryGoals• Represent the cosine of the acute angle of a right triangle as the ratio of the lengths of the adjacent side and the hypotenuse.

The cosine of the angle a, cos(a), is the ratio of the length of the adjacent side to the length of the hypotenuse.

cos(a) =

1

10 cos(a) =

2

13 cos(a) =

2

18

TrigonometryGoals•Calculate the ratio of the adjacent side and the hypotenuse of a right triangle using the inverse cosine of the acute angle.

The inverse sine cos-1(a), is the answer to the question: if the ratio is known what is the angle?

71.56 =cos−1

1

10

⎛

⎝⎜⎞

⎠⎟ 56.30 =cos−1

2

13

⎛

⎝⎜⎞

⎠⎟ 45=cos−1

3

18

⎛

⎝⎜⎞

⎠⎟

TrigonometryGoals• Calculate the sine of the acute angle of a right triangle.• Calculate the cosine of the acute angle of a right triangle.

320.14 sin(42°) to the nearest ten-thousandth

320.20 cos(88°) to the nearest ten-thousandth

320.39 Calculate sin(25°) and sin(50°). Is the sine of the angle doubled if the angle is doubled?

HW due Thursday 1/10316.2, 3, 5, 9, 12, 13, 317.19, 21, 320.1, 3, 4, 5, 6, 8, 14, 20, 31, 32, 39, 322.43, 47, 50

TrigonometryApplications of the sine and cosine

• if the hypotenuse and the angle of elevation are known then the length of the opposite side can be determined.

A hot-air balloon is tied to the ground with two taut (straight) ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50° with the ground.

TrigonometryApplications of the sine and cosine

• if the hypotenuse and the angle of elevation are known then the length of the opposite side can be determined.

A hot-air balloon is tied to the ground with two taut (straight) ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50° with the ground. What is the balloon’s height?

Let x be the vertical height

sin(50°) =x

110 ftx=0.766 •110 ft=84.3 ft

TrigonometryApplications of the sine and cosine

• if the hypotenuse and the angle of elevation are known then the length of the adjacent side can be determined.

A hot-air balloon is tied to the ground with two taut (straight) ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50° with the ground. What is the distance along the ground between the two ropes?

TrigonometryApplications of the sine and cosine

• if the hypotenuse and the angle of elevation are known then the length of the opposite side can be determined.

A hot-air balloon is tied to the ground with two taut (straight) ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50° with the ground. What is the distance along the ground between the two ropes?

Let x be the horizontal distance

cos(50°) =x

110 ftx=0.643•110 ft=70.7 ft

TrigonometryGoals• Solve narrative problems by applying trigonometry of a right triangle.

HW due Tuesday 1/11325.6, 7, 9, 11, 14, 16, 326.19, 23, 26, 327.29, 30

TrigonometryGoals• Solve narrative problems by applying trigonometry of a right triangle.

HW due Wednesday 1/12329.13, 18, 19, 330.21, 24, 28, 29, 32