geometry final

7/28/2019 Geometry Final

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PLUMBING ARITHMETIC

GEOMETRY

PLANE GEOMETRY

I. DEFINITION

Geometry is the branch of mathematics which deals with the properties of shapes and spaces. The termgeometry was derived from the Greek words, ge meaning earth and metria meaning measurement.

II. ANGLES

Different types of angles:

Name MeasureNull or zero angle 0

Acute angle > 0 but < 90Right angle 90

Obtuse angle > 90 but < 180

Straight angle 180Reflex angle > 180 but


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Table showing the regular polygons:

Name Sides AngleEquilateraltriangle

3 60

Square 4 90

Regular pentagon 5 108Regular hexagon 6 120

Regular heptagon 7 128.57Regular octagon 8 135Regular nonagon 9 140

Regular decagon 10 144Regular hectagon 100 176.4

Regular megagon 106 179.99964Regulagoogolgon

100100 180

Polygons are names according to the number of sides or vertices.

Number of sides Sides

3 Triangle4 Quadrilateral

5 Pentagon6 Hexagon7 Heptagon

8 Octagon9 Nonagon

10 Decagon11 Undecagon

12 Dodecagon100 Hectagon

1000000 Megagon10100 Googolgon

A convex polygon is a polygon having each interior angle less than 180 while a concave polygon is a polygonhaving an interior angle grater than 180

Convex Polygon Concave Polygon

Reentrant angle is the inward-pointing angle (angle A in the figure) of the concave polygon while the other anglesare called salient angles.

Number of Diagonals of Polygon Sum of Interior angles

of interior angles = (n 2)180


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IV. QUADRILATERALS

Parallelogram

Past Board Exam:One side of a parallelogram is 10m and its diagonals are 16m and 24m, respectively. Its area is: Ans: 158.75 sq.m.

Rhombus

Past Board Exam:The area of a rhombus is 132 sq.m. It has one diagonal equal to 12m. Determine the length of the sides of the rhombus.Ans. 12.53 m

General Quadrilateral

Past Board Exam:The sides of a quadrilateral are 12m, 8m, 16.97m, and 20m, respectively. Two opposite interior angles have a sum of 225deg. Find the area of the quadrilateral. Ans. 168 sq.m.

Past Board Exam:Find the number of diagonals of dodecagon. Ans. 54

Past Board Exam:Find the measurement of each interior angle of a regular icosagon. Ans. 162

Past Board Exam:Find the area of a pentagon that is inscribed in a circle if the area of the circle is 314 m2. Ans. 237.64 m2


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Past Board Exam:A circle having an area of 224 m2 is inscribed in an octagon. Find the area of an octagon. Ans. 236.3 m2

Theorems on Circles

CROSS CHORD

( ) ( ) ( ) ( )

( )1

2AC BD

AE EB CE DE

Arc Arc

ADC ABC

BAD BCD

=

= +

= =

TANGENT-SECANT

( ) ( ) ( )

( )

2

1

2BC AC

OC OA OB

Arc Arc

=

=

PERIPHERAL ANGLE

Prism

Bh =


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The bases of a right prism are pentagons with each side 6 cm long. The bases are 12 cm apart. What is the volume of the

prism?a. 733.2 cm3 c. 713.12 cm3

b. 743.2 cm3 d. 723.2 cm3

Find the volume of a right circular cylinder whose lateral area is 25.918 m2 and base area of 7.068 m2.a. 19.44 m3 c. 20.53 m3

b. 15.69 m3 d. 18.12 m3

Truncated Prism

R ave

ave

A h

hh

n

=

=

. The

A truncated prism having a square base has a volume of 1,000 m3. The heights of the prism at each corner arerespectively 7m, 7m, 10m, and 10m. What is the area of the base?

a. 117.65 m2 c. 134.82 m2b. 92.12 m2 d. 125.12 m2

Pyramid

1

3Bh =

A regular triangular pyramid has an altitude of 9m and a volume of 187.06 m3. What is the base edge?a. 18 m c. 12 mb. 14 m d. 16 m

Determine the volume of a regular tetrahedron whose side is 3 m.a. 3.182 m3 c. 5.321 m3

b. 2.983 m3 d. 1.119 m3

Similar Figures

If the edge of a cube is increased by 30%, by how much is the surface area increased?


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a. 67% c. 69%b. 63% d. 65%

A right circular cone with an altitude of 8cm is divided into two segments. One is a smaller circular cone having the samevertex with volume equal to of the bigger one. Find the altitude of the smaller cone.

a. 4.52 cm c. 5.04 cmb. 6.74 cm d. 6.12 cm

Cone

21

3r h

LSA rL

=

=

A conical vessel has a height of 24cm and a base diameter of 12 cm. It holds water to depth of 18cm above its vertex.Find the volume of its content.

a. 387.4 cm3

c. 383.5 cm3

b. 381.7 cm3 d. 385.2 cm3

The slant height of a right circular cone is 5 m long. The base diameter is 6 m. What is the lateral area?a. 37.7 m2 c. 44.3 m2

b. 47.1 m2 d. 40.8 m2

A cone was formed by rolling a thin sheet of metal in the form of a sector of a circle 72 cm in diameter with a central angle210o. What is the volume of the cone?

a. 13,318 cm3 c. 13,716 cm3

b. 13,602 cm3 d. 13,504 cm3

Frustum

( )1 2 1 23

hA A A A = + +

An artificial lake, 5 m deep, is to be dug in the form of a frustum of an inverted pyramid. The level bottom is 8 m by 80 mand its top is 10 m by 100 m. How many cubic meters of earth are to be removed?

a. 4,067 m3 c. 4,286 m3

b. 4,417 m3 d. 4,636 m3

The lateral edge of the frustum of a regular pyramid is 1.80 m long. The upper base is a square 1m x 1m and the lowerbase 2.40m x 2.40m square. Determine the volume of the frustum.

a. 4.6 m3 c. 5.7 m3

b. 3.3 m3 d. 6.5 m3

Prismatoid


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A certain quartz crystal with plane surfaces has the dimensions shown in the figure. If the upper and lower bases lie inparallel planes, find the volume of the crystal.

a. 1,763 cm3 c. 1,904 cm3

b. 1802 cm3 d. 1844 cm3

Sphere

3

2

4

3

4

r

SA r

=

=

What is the surface area of a sphere whose volume is 36 m3?a. 52.7 m2 c. 46.6 m2

b. 48.7 m2 d. 54.6 m2

Spherical Segment/Zone

( )2

33

2zone

hr h

A rh

=

=

What is the area of the zone of a spherical segment having a volume of 1,470.265 m3 if the diameter of the sphere is30m?

a. 659.734 m2 c. 848.230 m2

b. 565.487 m2 d. 753.982 m2

A mixture compound from equal parts of two liquids, one white and the other black was placed in a hemispherical bowl.The total depth of the two liquids is 6. After standing for a short time, the mixture separated. The white liquid settledbelow the black. If the thickness of the segment of the black liquid is 2, find the radius of the bowl.

a. 7.53 in c. 7.73 inb. 7.33 in d. 7.93 in

Spherical Wedge/Lune


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Find the radius of the spherical wedge whose volume is 12 m3 with a central angle of 1.80 radians.

a. 2.36 m c. 2.52 mb. 2.73 m d. 2.15 m

Spherical Cone

22

3r h =

A spherical cone is made from a sphere having a radius of 5 m. If the vertex angle of the cone is 40o, find its volume.a. 15.71 m3 c. 14.61 m3

b. 13.42 m3 d. 16.02 m3

Spherical pyramid

( ) ( )

3

540

2 180

r E

E A B C D n

=

= + + +

The base angles of a quadrangular spherical pyramid are 95o, 108o, 128o and 162o. The diameter of the sphere is 24 cm.

What is the volume of the pyramid?a. 1,528 cm3 c. 1,714 cm3

b. 1,337 cm3 d. 1,904 cm3


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ANALYTIC GEOMETRY

I. INTRODUCTION

Analytic Geometry deals with geometric problems using coordinates system thereby converting it into algebraic

problems.

Rene Descartes (1596 1650, Cartesius in Latin language) is regarded as a founder of analytic geometry by

introducing coordinates system in 1637. Regular Coordinate System (Also known as Cartesian Coordinates

System).


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Point O is the origin and has coordinates (0, 0). The x-coordinate orabscissa is always measured from the y-axis

while the y-coordinate orordinate is always measured from the x-axis. The point P has 5 and 3 as abscissa and

ordinate, respectively.

II. DISTANCE BETWEEN TWO POINTS IN PLANE

Consider two points whose coordinates are (x1, y1) and (x2, y2), respectively. A right triangle is formed with the

distance between two points being the hypotenuse of the right triangle.

Using Pythagorean Theorem, the distance between two points can be calculated using:

( ) ( )2 2

2 1 2 1d x x y y= +

This formula is known as the distance formula.

III. DISTANCE BETWEEN TWO POINTS IN SPACE

Consider three axes namely, x, y, and z and two points with coordinates (x1, y1, z1) and (x2, y2, z2), respectively.


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( ) ( ) ( )2 2

2 1 2 1 2 1d x x y y z z = + +

IV. SLOPE OF A LINE (m)

Consider two points whose coordinates are (x1, y1) and x2, y2), respectively. A line is formed by connecting the two

points. The slope of the line is defined as the rise (vertical) per run (horizontal).

rise yslope m

run x

= = =

Where: denotes an increment

2 1

2 1

tany y

x x

=

But tan m = m = 2 1

2 1

y y

x x

A line parallel to the x-axis has a slope of zero while a line parallel to the y-axis has a slope of infinity ( ) .

For parallel lines with slope of m1 and m2, respectively, the slopes are the same.

1 2m m=

For the perpendicular lines with slopes of m1 and m2, respectively, the slope of one is the negative reciprocal of the

other.

2

1

1m

m=

V. ANGLE BETWEEN TWO LINES ( )

Consider two lines with slopes of m1 and m2.

The angle, between these lines (line 1 and line 2) may be calculated using the following formula:


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2 1

2 1

tan1

m m

m m

=

+1 2 1

2 1

tan1

m m

m m

= +

VI. ANGLE BETWEEN TWOLINES ( )

Consider a point with coordinates (x1, y1) and a line with equation Ax + By + C = 0.

1 1

2 2

Ax By Cd

A B

+ +=

+

Use

+ if B is positive and the point is above the line or to the right of the line

+ if B is negative and the point is below the line or to the left of the line

- If otherwise

VII. ANGLE BETWEEN TWO LINES ( )

Consider two parallel lines with equation as shown in the figure. The (perpendicular) distance, d, between the two

lines is:

1 2

2 2

C Cd

A B

=

+

Use the sign (either + or -) that would make the distance positive.

VIII. DIVISION OF LINE SEGMENT

Consider two points with coordinates (x1, y1)and (x2, y2). The line segment formed by these two points is divided by a

point P whose coordinates are (x, y).

Let r1 and r2 be the corresponding ratio of its length to the total distance between two points.


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The abscissa of the point, P is: ( ) ( )1 2 2 1

1 2

x r x rxr r

+=+

The ordinate of the point, P is:( ) ( )1 2 2 1

1 2

y r y ry

r r

+=

+

If the point, P is at the mid-point of the line segment, then the abscissa and ordinate of the point are the following:

1 2

2

x xx

+= and 2 1

2

y yy

+=

IX. AREA OF POLYGON BY USING THE COORDINATES OF ITS VERTICES

Consider a polygon whose vertices have coordinates of (x1, y1), (x2, y2) and (x3, y3)

The arrow shown in the figure moving in a counter clockwise direction indicates that the vertices must be written in the

equation below in counter clockwise direction.

31 2 1

1 2 3 1

1

2

xx x xA

y y y y=

( ) ( )1 2 2 3 3 1 1 2 2 3 3 11

2A x y x y x y y x y x y x= + + + +

X. EQUATION OF LINES

A line is defined as the shortest distance between two points. The following are the equations of the lines:

A. General Equation: Ax + By + C = 0

B. Point-Slope Form: y-y1 = mx(x x1)

C. Slope-Intercept Form: y = mx + b

D. Two-Point Form: ( )2 11 12 1

y yy y x x

x x

=


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E. Intercept Form: 1x y

a b+ =

XI. CONIC SECTIONS

Conic section (or simply Conic) is the locus of a point which moves so that its distance from a fixed point (focus) is in

constant ratio, e (eccentricity) to its distance from a fixed straight line (directrix).

The term conic was first introduced by a renowned mathematician and astronomer of antiquity, Apollonius (c.255

170 B.C.)

Also, the term conic section was due to the fact that the section is formed by a plane made to intersect a cone.

Circle is produced when the cutting plane is parallel to the base of the cone.

Ellipse is produced when the cutting plane is not parallel (or inclined) to the base of the cone.

Parabola is produced when the cutting plain is parallel to the element (or generatix) of the cone.

Hyperbola is produced when the cutting plane is parallel to the axis of the cone.

GENERAL EQUATION OF CONIC SECTION


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Ax2 + Bxy + Cy2 + Dx + Ey + F= 0

When B is not equal to zero, then the principal axes of the conic are inclined (not parallel to the coordinate axes). The

curve can be identified from the equation given by determining the value of the determinant, B2 4AC.

B2 4AC Conic Section Eccentricity

1.0

When B is equal to zero, then the principal axes of the conic are parallel to the coordinates axes (x and y axes). To

identify the curve, compare the coefficients of A and C.

If A = C, the conic is a circle.

If A C but the same signs, the conic is an ellipse.

If A and C have different signs, the conic is hyperbola.

If either A or C is zero, the conic is a parabola.

The conic sections have geometric properties that can be used for some engineering application such as beams ofsound and reflection of rays of light.

Circle reflects rays issued from the focus back to the center of the circle.

Parabola reflects rays issued from the focus as a parallel (with respect to its axis) outgoing beam.

Ellipse reflects rays issued from the focus into the other focus.

Hyperbola reflects rays issued from the focus as if coming from the other focus.

A. CIRCLES

A circle is a locus of a point that which moves so that it is equidistant from a fixed point called center.


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1. General Equation: x2 + y2 + Dx + Ey + F = 0

If D & E = 0, center is at the origin (0, 0)

If either D or E, or both D & E 0, the center is at (h, k).

2. Standard equations:

C (0, 0)

X2 + y2 = r2

C (h, k)

(x h)2 + (y k)2 = r2

When the equation given is general equation rather than standard equation, the center (h, k) of the circle and its

radius (r) can be determine by converting the general equation to the standard using the process known as

completing the square. Or using the following formulas:

General equation: Ax2 + Cy2+ Dx + Ey + F= 0

Center (h, k)2

Dh

A

=

2

Ek

A

=

Radius (r)2 2

2

4

4

D E AFr

A

+ =

B. PARABOLAS

A parabola is a locus of a point which moves so that it is always equidistant to a fixed point called focus and it a fixed

straight line called directrix.


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Let:

a = distance from vertex V to focus F

d = distance from point to directrix

f = focal distance

1. General equations:

A. Axis parallel to the y-axis

Ax2 +Dx + Ey + F = 0

B. Axis parallel to the x-axis:

Cy2 +Dx +Ey + F = 0

2. Standard Equations:

Vertex (V) at origin (0, 0)

A. Axis along x-axis:


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Opens to the right

Y2 = 4ax

Opens to the left

Y2 = -4ax

B. Axis along y-axis:

Opens up

X2 = 4ay

Opens down

X2 = -4ay

Vertex (V) at (h, k)

A. Axis parallel to the x-axis

Opens to the right


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(y-k)2 = 4a(x-h)

Opens to the left

(y-k)2 = -4a(x-h)

B. Axis parallel to the y-axis

Opens upward

(x-h)2 = 4a(y k)

Opens downward

(x-h)2 = -4a(y k)

The eccentricity of the parabola is the ratio of the distance to the focus to the distance to the directrix.


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fe

d= Since f = d, then: e = 1

The latus rectum of a parabola is a line that passes through the focus and perpendicular to the axis of the conic.

LR =4a

When the equation given is a general equation rather than standard equation, the vertex V(h, k) of the parabola and its

focal length or focal radius a can be calculated by converting the general equation to standard using the process known

as completing the square.

The following formulas can be obtained:

For axis horizontal: Cy2 +Dx +Ey + F = 0

24

4

E CFh

CD

=

2

Ek

C

=

4

Da

C

=

For axis vertical: Ax2 +Dx + Ey + F = 0

2

Dh A

=

2

4

4D AFk

AE= 4E

a A

=

C. ELLIPSES

An ellipse is locus of a point which moves so that the sum of its distance to the fixed points (foci) is constant and is equal

to the length of the major axis (2a).

1. General Equation: Ax2 + Cy2+ Dx + Ey + F= 0

Note: d1 + d2 = 2a. The maor axis = 2a, is the distance from V1 to V2.

When the point is located along the minor axis as shown in the following figure:

b2 + c2 = a2


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The relationship between a, b and c is with a is always greater than b.

If D & E = 0, center is at the origin (0, 0). If either D or E, of both D & E 0, the center is at (h, k).

2. Standard Equations:

Center, C at (0, 0)

Major axis is horizontal:

2 2

2 21

x y

a b+ =

Major axis is vertical:

2 2

2 21

x y

b a+ =

Center, C at (h, k)

Major axis is horizontal:

( ) ( )2 2

2 21

x h y k

a b

+ =


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Major axis is vertical:

( ) ( )2 2

2 21

x h y k

b a

+ =

The eccentricity of an ellipse is the ratio of the distance to the directrix.

fe

d=

When the point P(x, y) is the minor axis:


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ae

D=

If f = c, then eccentricity of an ellipse, e isc

ea

=

Since a < D and c < a, then the eccentricity of an ellipse is always less than 1. e < 1

The latus rectum of an ellipse is a line that passes through the focus and perpendicular to the axis of the conic.

22b

LR a= where: a = semi-major axis; b = semi-minor axis

When the equation given is a general equation rather than standard equation, the center (h, k) of an ellipse and its

focal length c can be calculated by converting the general equation to standard using the process known as

completing the square. The following formulas can be obtained:

3. General equation: Ax2 + Cy2+ Dx + Ey + F= 0

2

Dh

A

= 2

Ek

C

=

2 2c a b=

D. HYPERBOLAS

A hyperbola is a locus of a point which moves so that the difference of the distance to the fixed points (foci) is

constant and is equal to the length of the transverse axis (2a).


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1. General equations:

A. Transverse axis horizontal

Ax2 - Cy2+ Dx + Ey + F= 0

Transverse axis is the axis that passes through the foci, vertices and the center of the hyperbola while the

conjugate axis is the one that is perpendicular to the transverse axis.

Length of the transverse axis = 2a = 2 C

Length of the conjugate axis = 2b = 2 A

Where: A and C are the numerical coefficients (absolute value) of x2 and y2, respectively.

The relationship between a, b and c is a2 + b2 = c2

B. Transverse axis vertical:

Cy2- Ax2 + Dx + Ey + F= 0

Length of the transverse axis = 2a = 2 A

Length of the conjugate axis = 2b = 2 C


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Where: A and C are the numerical coefficients (absolute value) of x2 and y2, respectively.

Also, the relationship between a, b and c is a2 + b2 = c2

2. Standard equations:

Center, C at (0, 0)

Transverse axis horizontal:

2 2

2 21

x y

a b =

Transverse axis vertical:

2 2

2 21

y x

a b =

Center, C is at (h, k)

Transverse axis horizontal:


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( ) ( )2 2

2 21

x h y k

a b

=

Transverse axis vertical:

( ) ( )2 2

2 21

y k x h

a b

=

The eccentricity of a hyperbola is the ratio of the distance to the focus to the distance to the directrix.

ce

a=

ae

D=

Since a> c and D > a, then the eccentricity of a hyperbola is always greater than 1. e > 1


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The latus rectum of an ellipse is a line that passes through the focus and perpendicular to the axis of the conic

22b

LRa

= where: a = semi-major axis

b = semi-minor axis

When the equation given is a general equation rather than standard equation, the center (h, k) of a hyperbola can

be calculated by converting the general equation to standard using the process known as completing thesquare.

The following formulas can be obtained:

2

Dh

A

= 2

Ek

C

=

XII. POLAR COORDINATES

Polar coordinates refers to the coordinates of a point in a system of coordinates where the position of a

point id determined by the length of ray segment (the radius vector) from a fixed origin (the pole) and the angle (the

polar angle) the ray (the vector)makes with a fixed line (the polar axis).

),r


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Polar angle is sometimes called the vectoral angle, the argument, the amplitude, or the azimuth of a point.

Relationships between polar coordinates and rectangular coordinates:

cosx r = siny r = 2 2r x y= +arctan

y

x =

geometry final

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