TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINESAyala Blvd., Ermita, Manila

www.tup.edu.ph

Chapter 1:

PLANE FIGURES

Landrito, Christopher Louie O. Prof. Nick Sarte ECET-2D

1.1 POLYGONS

Definition of PolygonA polygon is a closed plane figure formed by line

segments.

Parts of a Polygon:1.The side or the edge of a polygon is one of the line

segments that make up the polygon. Adjacent sides are pair of sides that share a common endpoint.

2.The vertices of a polygon are the endpoints of each side of the polygon. Adjacent vertices are endpoints of the side.

3.A diagonal of a polygon is a line segment joining two non-adjacent vertices of the polygon.

4.An interior angle is the angle formed by two adjacent sides inside the polygon.

5.An exterior angle is an angle that is adjacent to and supplementary to an interior angle of the polygon.

Side or edge

Interior angle

Diagonal

Vertex

Exterior Angle

A polygon may also be defined as a union of line segments such that:

i) Each vertex is a common endpoint of two adjacent line segments;

ii) No two adjacent line segments intersects except at an endpoint; and

iii) No two segments with the same endpoint are collinear.

Types of Polygons

1. Equiangular PolygonA polygon is equiangular if all of its angles are congruent.

2. Equilateral PolygonA polygon is equilateral if all of its sides are equal.

3. Regular PolygonRegular polygons are both equiangular and equilateral.

4. Irregular PolygonA polygon that is neither equiangular nor equilateral is sais to be an irregular polygon.

5. Convex PolygonEvery interior angle of a convex

polygon is less than 180’. If a line is drawn through the convex polygon, the line will intersect at mosttwo sides.

6. Concave PolygonA concave polygon has at least

one interior angle that measuresmore than 180’. If a line is drawnthrough a concave polygon, the line may intersect more than two sides.

Naming PolygonsPolygons are named according to

their number and sides. Generally, a polygon with n sides is called an n-gon . To form the name of polygons with 13 to 99 sides, begin with the prefix for the tens digit, followed by Kai (the Greek word for and) and the prefix for the units digit.

Number of Sides Name of Polygon

N N-gon

3 Triangle or trigon

4 Quadrilateral or tetragon

5 Pentagon

6 Hexagon

7 Heptagon

Number of Sides Name of Polygon8 Octagon

9 Nonagon or enneagon

10 Decagon

11 Undecagon or hendecagon

12 Dodecagon

13 Tridecagon or triskaidecagon

14 Tetradecagon or tetrakaidecagon

15Pentadecagon or pentakaidecagon

16Hexadecagon or hexakaidecagon

17Heptadecagon or heptakaidecagon

18 Octadecagon or octakaidecagon

19 Enneadecagon or enneakaidecagon

20 Icosagon

30 triacontagon

Number of Sides Name of Polygon

40 Tetracontagon

50 Pentacontagon

60 Hexacontagon

70 Heptacontagon

80 Octacontagon

90 Enneacontagon

100 Hectogon or hecatontagon

1,000 Chiliagon

10,000 Myriagon

10-6 Megagon

10100 Googolgon

Sides PrefixAnd

Sides(Ones Digit)

Suffix

20 Icosi or Icosa 1 Henagon

30 Triaconta 2 Digon

40 Tetraconta 3 Trigon

50 Pentaconta 4 Tetragon

60 Hexaconta Kai + 5 Pentagon

70 Heptaconta 6 Hexagon

80 Octaconta 7 Heptagon

90 Enneaconta 8 Octagon

9 Enneagon

For numbers from 100 to 999, form the name of the polygon by starting with the prefix for the hundreds digits taken from the ones digit taken from the ones digit, affix the word hecta, then follow the rule on naming polygons with 3 to 99 sides. However, one may use the form n-gon as in 24-gon for a polygon with 24 sides, instead of using the above method.

Examples:

1. A 54-sided polygon is called a pentacontakaitetragon.

2. A 532-sided polygon is called pentahectatriacontakaidigon.

50 and 4

Pentaconta Kai tetragon

500 30 And 2

Pentahecta Triaconta Kai Digon

60 and 9

Hexaconta Kai enneagon

3. A 60-sided polygon is called a Hexacontakaienneagon.

4. A 1069-sided polygon is called Chiliahexacontakaienneagon.

1000 60 And 9

Chilia Hexaconta Kai enneagon

Similar PolygonsThe ratio of two quantities is the

quotient of one quantity. Note, however, that the two quantities must be of the same kind. For example, the ratio of the measure of a side and an interior angle is meaningless because they are not quantities of the same kind. A proportion is an expression of equality between two ratios. That is, if two ratios a:b and c:d are equal, then the equation

is a proportion. Thus, you can say that a and b are proportional to c and d.

Two polygons are similar if their corresponding interior angles are congruent and their corresponding sides are proportional. Similar polygons have the same shape but may differ in size.

Consider the given similar polygons below:

The following relations between the two polygons are obtained using the concept of ratio and proportion:

1. The ratio of any two corresponding sides of similar polygons are equal.

Y1

X1

A1

Y2

X1

A1

2. The ratio of the areas of similar polygons is the

square of the ratio of any two corresponding sides.

3. The ratio of the perimeters of similar polygons is equal to the ratio of any two corresponding sides.

2

Properties of a Regular Polygon

A regular polygon of n sides can be subdivided into n congruent Isosceles triangles, whose base is a side of the polygon. The common vertex of these triangles is the center of the polygon.

a

.s

Perimeter

P=ns,

To find the perimeter of a polygon, add the lengths of the sides of the polygon. Since regular polygons are equilateral, the formula in finding the perimeter of a regular polygon is

where n is the number of sides and s is the length of each side.

The angle that is opposite a side of a regular polygon is a central angle of the polygon. It is the angle formed by two lines drawn from the center of the polygon to two adjacent vertices. Regular polygons are equiangular. Thus, the measure of each central angle is given by

Central Angle

Apothem

The altitude of the isosceles triangles that can be formed from a regular polygon is the apothem of the regular polygon. The apothem bisects the central angle and its opposite side. Thus, we can compute for the apothem as follows:

Solve for a,

Interior Angle

In each isosceles triangle, the measure of the base angles can be denoted by , and each interior angle of the regular polygon by 2 . Thus, the measure of each interior angle is solved as follows:

thus,

Sum of Interior Angle

Since the number of sides equals the number of interior angles, then the sum of interior angles is n times the measure of the interior angle. Hence,

AreaArea is the amount of two-

dimensional space that a plane figure occupies. To get the are of a regular polygon, multiply the area of the isosceles triangle by the number of triangles form, or .

Note that this is the same as one-half of the product of its perimeter and its apothem. In general, the formula for area of a regular polygon is given by

The formula for area of a regular polygon can be expressed in terms of its number of sides and the measure of one sides as follows:

DiagonalFrom any given vertex of a regular polygon, a diagonal is drawn from the vertex to a non-adjacent vertex. This means that you can construct a diagonal from each vertex of a polygon with n sides in n – 3 ways. Since there a n vertices and each diagonal has two endpoints, you can do this in only ways. Thus, the total number of the distinct diagonals of a regular polygon is

Example:

Find the area of a regular nonagon whose sides measure 3 units. Determine the number of distinct diagonals that can be drawn from each vertex and the sum of its interior angles.

Solution:A nonagon is 9-sided polygon.

Thus, n=9, s=3. Solve as follows:

Area of the polygon:

Number of Diagonals:

Sum of interior angles:

1.1 Exercises 1. Use the diagram below to answer the questions

(a) to (d).

a) Is the polygon convex or concave?b) How many diagonals can be drawn from

vertex A?c) How many sides does the polygon have?d) What is the name of the polygon?

A

2. Use the diagram below to answer questions (a) to (d).

a) Is the polygon convex or concave?b) How many diagonals can be drawn from

vertex A?c) How many sides does the polygon have?d) What is the name of the polygon?

A

3. What is the measure of an interior angle of a regular pencontakaitrigon?

5. Name each polygon with the given number of sides.a. 39b.127c. 821d. 86

6. The number of diagonals of a regular polygon is 35. Find the area of the polygon if its perimeter is 45 centimeters.

7. Find the number of diagonals of a regular polygon whose interior angle measures .

8. The ratio of areas between two similar triangles is 1:4. If one side of the smaller triangle is 2 units, find the measure of the corresponding side of the other triangle.

9. The sum of the interior angles of a regular polygon is . Find the area of the polygon if it’s perimeter is 45 centimeters.

10. Find the measure of an interior angle of a regular tridecagon.

Goodluck :”)

1.2 Triangles

The most fundamental subset of polygon is the set of Triangles.

Classification of Triangles according to Sides

1. Equilateral – a triangle with three congruent sides and three congruent angles. Each angle measures

2. Isosceles – a triangle with two congruent sides and two congruent angles.

3. Scalene - a triangle with no congruent sides and no congruent angles.

Scalene

Isosceles

Equilateral

Classification of Triangles according to Angles

1. Right – a triangle with a right angle ( ).

2. Oblique – a triangle with no right angle.

a) Acute – a triangle with three acute angles(less than ).

b) Equiangular – a triangle with three congruent angles. Each angle measures .

c) Obtuse – a triangle with one obtuse angle (more than but less than ) and two

acute angles.

Congruent Triangles

The word congruent is derived from the Latin word, congruere, which means agree. Two triangles are congruent when they have the same shape and size. Congruent triangles can be made to coincide part by part. Corresponding parts of congruent triangles are congruent. The symbol for congruent is .

Similar Triangles

Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. Similar triangles have the same shape but differ in size. Look at the similar triangles below.

Since the two triangles are similar, then the relations that exist between two similar polygons also hold. Thus, it follows that:

Parts of a TriangleA triangle has three possible bases and

three possible vertices. Any of the three sides of the triangles may be considered as the base of the triangle. The angle opposite the base is called vertex angle. The two angles adjacent to the base are called base angles.

a line segment drawn from a vertex perpendicular to the opposite side is called altitude. The point of intersection of the altitudes of a triangle is called orthocenter. A median of a triangle is the line segment connecting the midpoint of a side and the opposite vertex. The centroid is the point of intersection of the medians of the triangle.

An angle bisector divides an angle of the triangle into two congruent angles and has endpoints on a vertex and the opposite side. The point of intersection of the angle bisectors of a triangle is called incenter.

A perpendicular bisector of a side of a triangle divides the side into two congruent segments and is perpendicular to the side. The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle. The Euler line is the line which contains orthocenter, centroid, and circumcenter of a triangle. The centroid is located between the orthocenter and the circumcenter. However, in an equilateral triangle, the centroid, circumcenter, incircle and orthocenter are coincident.

Properties of Triangle Centers1. Orthocenter – is not always in the interior of the triangle. In

an obtuse triangle, the two sides of the obtuse angle and the corresponding altitudes are extended to meet a point outside the triangle. In the right right triangle, the orthocenter is on a vertex of the triangle.

2. Centroid – known as the center of mass of the triangle. Unlike the orthocenter, the centroid is always inside the triangle and for right, isosceles and equilateral triangles, the centroid is located one-third of the altitude from the base.

3. Incenter – is the center of the largest circle that can be inscribed in a triangle.

4. Circumcenter – is the center of the circle circumscribing a triangle. It is not always inside the triangle. The vertices of the triangle lie on the the circle and are equidistant from the circumcenter.

Consider an arbitrary triangle with sides a, b, and c, and angles A, B, and C. Let be the lenghts of the altitude, median, and angle bisector from vertex C, respectively. Then,

Altitude:

Altitude, Median, and Angle Bisector Formulas

C

ab

A Bc

Where s is the semi-perimeter of the triangle and

Medium:

Angle Bisector:

C

ab

A Bc

ab

A B

c

C

Facts about Triangles1. The sum of the lengths of any two sides of a

triangle is always greater than the third side. The difference between the lengths of any two sides is always less than the third side of a triangle.

2. The sum of the measures of the interior angles of a triangle is .

3. Two equiangular triangles are similar.4. Two triangles are similar if their corresponding

sides are parallel. Two triangles are similar if their corresponding sides are perpendicular.

5. In any right triangle, the longest side opposite the right angle is called hypotenuse.

6. If any two sides of a right triangle are given, the third side can be obtained by the Pythagorean theorem .

7. Two triangles are equal if the measure of the two sides and included angle of one triangle are equal to the measure of the two sides and the included angle of other triangle.

8. The line segment which join the midpoints of two sides of a triangle is parallel to the third side and equal to one-half the length of the third side.

9. In any triangle, the longest side is the opposite the largest angle, and the shortest side is opposite to the smallest angle.

10. The altitude h to the hypotenuse of the c of a right triangle divides the triangle into similar

Triangles. Each of the triangles formed by this altitude is similar to the original triangle.

11. Each leg of a right triangle is the geometric mean between the hypotenuse and the projection of the leg on the hypotenuse.

h

x

c-x

Formulas for the area of a

Triangle

In general, the area of the triangle is one half the product of its base and altitude.

To solve for the area of a triangle given the measures of two sides and an included angle, use the SAS formula.

SAS (Side – Angle – Side) Formula:

The area of a triangle is one-half the product of any two sides and he sine of their included angle.

When the measures of the three sides of triangle are given, the area of the triangle is determined by Heron’s formula.

b

a

Heron’s Formula or SSS (Three sides) Formula:

Where is called semi-perimeter.

B

A C

c a

b

1.2 ExamplesProblem:

1. The measure of the three sides of a triangle are AB = 30 in., and BC = 60 in. from a point D on side AB, a line DE is drawn through a point E on side AC such that angle AED is equal to angle ABC. If the perimeter of the triangle ADE is equal to 56 in., find the sum of the lengths of line segment BD and CE.

Solution: Draw the figure and label the parts with the

given measure.

The perimeter of triangle ABC is: P = 30+60+50=140 in. Notice that triangle ABC and triangle ADE are similar triangles since

Thus, the ratio

30

D

A E

50

C

60

of their perimeters is equal to the ratio of any of their corresponding sides.

Hence, BD+EC = 10+38 = 48 in.

BD = 30 – AD = 30 – 20 = 10

EC = 50 – AE = 50 – 12 = 38

2. If one side of a triangle is 20 units and the perimeter is 72 units, what is the maximum area that the triangle can have?

Solution:Imagine the side of length 20 units as the base of the triangle. Thus, the sum of the lengths of the order two-sides is P – 20= 52 units. Since the area of the triangle is maximum when the height is also maximum, the triangle is isosceles and the two sides measure 26 units each. By Pythagorean theorem,

Hence, the area is

26 26

1010

h

3. Derive formulas for the height and area of an equilateral triangle with side s.

Solution:In an equilateral triangle, the altitude divides the triangle into two congruent right triangles. Thus, by the Pythagorean Theorem.

h

s

Since each interior angle measures , use the SAS formula to find the area of the triangle.

1. Find the altitude and the area of an equilateral triangle whose side is 8cm long.

2. Is it possible to form a triangle with sides 2,4 and 8 units? Justify your answer.

3. The ratio of the base of an isosceles triangle to its altitude is 3:4. Find the measures of the angles of the triangle.

4. In an acute triangle ABC, an altitude AD is drawn. Find the area of triangle ABC if AB = 15 in.,AC = 18 in., and BD = 10 in.

5. Two altitudes of an isosceles triangle are equal to 20 cm and 30 cm. Determine the possible measures of the base angles of the triangle.

1.2 Exercises

6. In a right triangle, the bisector of the right angle divides the hypotenuse in the ratio 3 is to 50. Determine the measures of the acute angle of the triangle.

7. In a right triangle, a line perpendicular to the hypotenuse drawn from the of one of the sides divides the hypotenuse into segments which are 10 cm and 6 cm long. Find the lengths of the two sides of the triangles.

8. Is it possible to form a triangle with sides 20, 30 and 50 units? Explain.

9. Find the area of an equilateral triangle if its altitude is 5cm.

10. Derive the formula for the median of triangle ABC drawn from vertex C to side AB using the cosine law.

A quadrilateral , also known as tetragon or quadrangle, is a general term for a four-sided polygon.

The common parts of a quadrilateral are described as follows:

1. Side – is a line segments which joins any two adjacent vertices.2. Interior angle – is the angle formed between two adjacent sides.3. Height and altitude – It is the distance between two parallel sides of a quadrilateral. 4. Base - This is the side that is perpendicular to the altitude.5. Diagonal – This is the line segment joining any two non – adjacent vertices.

1.3 Quadrilateral

Classification of QuadrilateralsThe classification of quadrilaterals is based on the number of pairs of its parallel sides as shown in the figure below.

Quadrilateral

ParallelogramRectangle

Rhombus

Trapezoid

Square

Trapezium

Parallelogram has two pairs of parallel

sides.

Trapezoid has only one pair of parallel

sides.

Trapezium does not have any pair of

parallel sides.

Rectangle, Rhombus, and Square are

special types of parallelograms.

General Formulas for the Area of Quadrilaterals

Consider the quadrilateral below.

There are several useful formulas for the area of a plane convex quadrilateral in terms of sides a, b, c and d, and diagonal lengths and .

A

a b

d

D

c

B

C

Among them are the following:

Formula 1:

Where is the angle formed between and .

Formula 2:

Where the four sides are labeled such that

Formula 3:

ParallelogramA parallelogram is a quadrilateral in which

the opposite sides are parallel. The figure below illustrates an example of parallelogram.

A

C

B

D

h (height)

b (base)

Parallelograms have the following important properties:

1. Opposite sides are equal.2 Opposite interior angles are congruent.3. Adjacent angles are supplementary.4. A diagonal divides the parallelogram into two congruent triangles.5. The two diagonals bisect each other.

Diagonals of a Parallelogram If sides a and b, and the angle are given, then by the Cosine law, the diagonal may be obtained by the equation.

If any two parts of a parallelogram are given, the relationship among a, h and may be obtained from the right triangle. Using the other angle , the second diagonal maybe obtained by the same formula.

Perimeter of a

Parallelogram

Opposite sides of a parallelogram are equal. Thus, its perimeter is given by

Area of a Parallelogram

The area of a parallelogram can be obtained by any of the following formulas:

Formula 1:

where b is the length of the base, and h is the height.

Formula 2:

where a and b are the lengths of the sides of the parallelogram a is any is any inetrior angle.

A = bh,

In problems involving area of a parallelogram, one will have to find the height h, or the angle using the right triangle containing the parts a, h, and .Since a diagonal divides the parallelogram into two congruent triangles, the area of the parallelogram is twice the area of one of the two triangles. Thus, if two sides a and b, and an angle are given you can obtained the area using SAS formula. The area of the parallelogram is determined by

Note that the angle is any interior angle since

Rectangle

A rectangle is essentially a parallelogram in which the interior angles are all right angles.

A

h

B

CDb

Diagonals of a RectangleA diagonal of a rectangle cuts the rectangle into two congruent right triangles. The diagonal AC divides the rectangle ABCD into congruent right triangles ADC and ABC. Since the diagonal of the rectangle forms right triangles that include the diagonal and two sides of the rectangle, one can always compute for the third side with the use of the Pythagorean Theorem, if any two of these parts are given. Thus the diagonal d = AC may be determined using the equation

Perimeter of a RectangleThe perimeter is the sum of the four sides. Thus,

P = 2b + 2h

Area of a RectangleIf b is the length of the base and h is the

height, then the formula for the area of a rectangle is

A = bh

Square

A square is a special type of a rectangle in which all the sides are equal. Since all sides and interior angles are equal, a square is classified as a regular polygon of four sides.

a

a

d

Diagonal of a SquareThe steps in finding the diagonal of a

square is similar to the steps used in a rectangle. Thus,

where a is the length of one side of a square.

Note that, if the length of the diagonal is given one can always compute for the length of the sides of the square using the same formula.

Area of Square

The formula for the area of a square is given by

Rhombus

A rhombus is a parallelogram in which all sides are equal.

A rhombus may also be defined as an equilateral parallelogram.The terms “rhomb” and “diamond” are sometimes used instead of rhombus. A rhombus with an interior Angle is sometimes called lozenge.

h

The Diagonal of a Rhombus

Just like the square, the diagonals of a rhombus are perpendicular bisectors. Thus the angle formed by the diagonal ,measure and the length of each side of the rhombus is given by

Also, the diagonals of the rhombus are angle bisectors of the vertices. By the Cosine Law, the diagonals may be obtained in a similar manner like that of parallelogram.

Thus,

and

One can also verify that the angle opposite the shorter diagonal may be obtained by the formula

Where d is the longer diagonal and is the angle opposite the shorter diagonal.

The Perimeter of a Rhombus

If b is the measure of one side of a rhombus, then the perimeter is given by

Area of a RhombusThe area of a rhombus may be determined by any of the following ways:

The area is one half the product of its two diagonals.

Since a rhombus is a parallelogram, the area is also the product of the base times the height.

A= bh

The area is twice the area of one of two congruent triangles formed by one of its diagonals. This is the same method used in finding the area of a parallelogram.

TrapezoidA trapezoid is a quadrilateral with one pair of parallel sides.

• If the non-parallel sides are congruent, the trapezoid is called an Isosceles Trapezoid.

• A trapezoid which contains two right angles is called a right trapezoid.

h

a

b

h

Area of Trapezoid

The area of a trapezoid is equal to the product of the mean of the bases and the height. In symbols, the areas is given by the formula.

The median of a trapezoid is the line segment parallel to and midway between the bases of the trapezoid. Thus, and A = mh.

Trapezium

A trapezium is a quadrilateral with no parallel sides. In finding the area of the trapezium you may use the three formulas for the area of the quadrilateral.

Problem:

1. Find the area and the perimeter of the right trapezoid shown in the figure.

1.3 Examples

8

11

Solution:

To find the area, The length of the base and the height of this triangle are 3 and h, respectively, where:

thus the area and the perimeter of the trapezoid are:

2. If ABCD is a rhombus, AC=4, and ADC is an equilateral triangle, what is the area of rhombus?

Solution:A B

CD

3. Find the area of a parallelogram with a base of 12 centimeters and a height of 5 centimeters.

Solution: 

A= bhA= (12 cm) · (5 cm)A= 60 cm2

1. Determine the area of a rectangle whose diagonal is 24 cm and the angle between the diagonals is 60 degree.

2. A garden plot is to contain 240 sq. ft. if its length is to be three times its width , what should its dimensions be?

3. A rhombus has diagonals of 32 and 20 inches. Find the area and the angle opposite longer the diagonal.

1.3 Exercises

4. If you double the length of the side of the square by how much do you increase the area of the square?

5. Find the area and perimeter of the square whose diagonal is 15 units long.

6. The vertical end of a trough, which is the form of a trapezoid, has the following dimensions: width at the top is 1.65 m, width at the bottom is 1.15 m, and depth is 1.35 m. Find the area of this section of the trough.

7. If the diagonal length of the square is tripled, how much is the increase in the perimeter of the square.

8. If the length and width of the rectangle are

doubled, by what factor is the length of its

diagonal multiplied?

9.Find the height of a parallelogram with sides

10 and 20 inches long, and an included

angle of 35 degree. Also calculate the area

of the figure.

10. Find the area and perimeter of a square

whose diagonal is 20 units long.

The End :) Prepared by:

Landrito, Christopher Louie O.ECET-2D