Factorising Expressions Presentation in Green Brown Illustrative Scrapbook Style.pdf

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1. TRIANGLES AND
RECTANGLES BY GROUP 1EDP CODE
55269 MATH 122 MEMBERS: E/C AYHON, LORENZE E/C MONSANTO, JHON MIKE E/C DACALOS, JAMES LOUIE E/C GENOBIA, JOHN ROMUEL E/C ABAN, CHRISTIAN LARRY E/C BUSTAMANTE, STEPHEN E/C SELLEZA, SHRIEP MAVERICK E/C TAMAYO, IAN

2. Learning objectiveDefine triangles and rectangles, including
their properties and classifications
Apply the formulas to some problems on
triangle & rectangle
Recognize and explain the real-world
applications of triangles and rectangles
in maritime field. E/C AYHON, LORENZE After the discussion the students will be able to:

3. hISTORY OF TRIANGLE
Ancient Era
2500 BCE – Egyptian Pyramids
Triangular geometry applied in pyramid
construction, showing early knowledge of
slope and stability.
2000 BCE – Heron’s Formula (Greece)
First known formula for finding the area
of a triangle using only side lengths.E/C AYHON, LORENZE

4. TRIANGLE Classical Era
600 BCE – Pythagoras (Greece)
Introduced the Pythagorean Theorem a² + b² = c²,
transforming triangle study.
300 BCE – Euclid’s Elements
Systematized triangle properties, angles, and
congruence in a foundational geometry text.
140 BCE – Hipparchus (Greece)
Developed early trigonometry, applying triangle
ratios to astronomy. hISTORY OF TRIANGLEE/C AYHON, LORENZE

5. TRIANGLE Medieval Era
500–1500 CE – Islamic Golden Age
Scholars like Al-Biruni and Al-Kashi advanced
trigonometry for celestial navigation.
Christian Iconography
Triangle used to symbolize the Holy Trinity (Father,
Son, Holy Spirit).
Jewish Symbolism
Star of David, composed of two interlaced triangles,
represented duality and balance. hISTORY OF TRIANGLEE/C AYHON, LORENZE

6. TRIANGLE Modern Era
1800s–1900s – Structural Engineering
Triangles used in bridges, trusses, and ship hulls for
stability and load distribution.
20th Century – Symbolism and Psychology
Triangle symbolized hierarchy (Maslow’s pyramid) and
mind-body-spirit unity.
Present Day – Digital Applications
Triangles form the basis of computer graphics, 3D
modeling, architecture, and robotics. hISTORY OF TRIANGLEE/C AYHON, LORENZE

7. hISTORY OF RECTANGLE RECTANGLE Ancient Era
3000 BCE – Mesopotamian Land Measurement
Rectangles used in surveying and land division; recorded on clay
tablets (e.g., Plimpton 322).
2500 BCE – Egyptian Architecture
Rectangular layouts applied in temples, homes, and rope-
stretching methods to form right angles.
1800 BCE – Moscow Mathematical Papyrus (Egypt)
Contained problems on rectangular areas, reflecting practical
geometry in agriculture and construction.
500 BCE – Chinese Nine Chapters on the Mathematical Art
Introduced area and volume formulas for rectangles, applied in
farming and civil engineering. E/C AYHON, LORENZE

8. RECTANGLE Medieval Era
800–1400 CE – Islamic Architecture
Courtyard designs and geometric tiling
emphasized rectangular symmetry and
proportion.
1200 CE – European Manuscripts
Rectangular grids applied in illuminated texts,
mapping, and early cartography.hISTORY OF RECTANGLE E/C AYHON, LORENZE

9. RECTANGLE Modern Era
1800s–1900s – Industrial Design
Rectangles standardized in machinery,
packaging, and modular architecture for
efficiency.
20th Century – Digital and Graphic Design
Rectangles formed the basis of screens,
layouts, and pixel grids in computing.hISTORY OF RECTANGLE E/C AYHON, LORENZE

10. A triangle is a type of polygon, which has three
sides, and the two sides are joined end to end is
called the vertex of the triangle. An angle is formed
between two sides. This is one of the important
parts of geometry.
Some major concepts, such as Pythagoras theorem
and trigonometry, are dependent on triangle
properties. A triangle has different types based on
its angles and sides.
DEFINITION OF TRIANGLEE/C AYHON, LORENZE

11. TYPES OF TRIANGLE
On the basis of length of the
sides, triangles are classified into
three categories:
⦁ Scalene Triangle
⦁ Isosceles Triangle
⦁ Equilateral Triangle E/C AYHON, LORENZE

12. A scalene triangle is a type of triangle,
in which all the three sides have
different side measures. Due to this,
the three angles are also different
from each other. SCALENE TRIANGLEE/C AYHON, LORENZE

13. In an isosceles triangle, two sides
have equal length. The two angles
opposite to the two equal sides are
also equal to each other. ISOSCELES TRIANGLEE/C AYHON, LORENZE

14. An equilateral triangle has all three
sides equal to each other. Due to this
all the internal angles are of equal
degrees, i.e. each of the angles is 60° EQUILATERAL TRIANGLEE/C AYHON, LORENZE

15. TYPES OF TRIANGLE
On the basis of measurement of the angles,
triangles are classified into three
categories:
Acute Angle Triangle
Right Angle Triangle
Obtuse Angle TriangleE/C AYHON, LORENZE

16. ACUTE ANGLED TRIANGLEAn acute triangle has all of its
angles less than 90°. E/C AYHON, LORENZE

17. An obtuse triangle has
any of its one angles
more than 90°. OBTUSE ANGLED TRIANGLEE/C AYHON, LORENZE

18. RIGHT ANGLED TRIANGLEIn a right triangle, one of the
angles is equal to 90° or right
angle. E/C AYHON, LORENZE

19. A rectangle is a closed 2-D shape, having 4 sides, 4
corners, and 4 right angles (90°). The opposite sides
of a rectangle are equal and parallel. Since, a
rectangle is a 2-D shape, it is characterized by two
dimensions, length, and width. Length is the longer
side of the rectangle and width is the shorter side. DEFINITION OF RECTANGLEE/C AYHON, LORENZE

20. ⦁ Since, all the angles of a rectangle are equal, we
also call it an equiangular quadrilateral. A
quadrilateral is a 4-sided closed shape.
⦁ Since, a rectangle has parallel sides, we can also
call it a right-angled parallelogram. A parallelogram
is a quadrilateral whose opposite sides are equal
and parallel. Rectangles are special case of
paralleograms. What Else Can We Call a Rectangle? E/C AYHON, LORENZE

21. PRESENTATION OF FORMULAS AND
PROPERTIES Formulas for triangle: E/C DACALOS,
JAMES LOUIE

22. PRESENTATION OF FORMULAS AND
PROPERTIES Formulas for triangle: E/C DACALOS,
JAMES LOUIE

23. PRESENTATION OF FORMULAS AND
PROPERTIES Formulas for triangle: E/C DACALOS,
JAMES LOUIE

24. PRESENTATION OF FORMULAS AND
PROPERTIES Formulas for rectangle: E/C DACALOS,
JAMES LOUIE

25. PRESENTATION OF FORMULAS AND
PROPERTIES Triangle Properties
* A triangle has three sides and three angles.
* The sum of the angles of a triangle is always 180 degrees.
* The exterior angles of a triangle always add up to 360 degrees.
* The sum of consecutive interior and exterior angle is supplementary.
* The sum of the lengths of any two sides of a triangle is greater than the length of the
third side. Similarly, the difference between the lengths of any two sides of a triangle is
less than the length of the third side.
* The shortest side is always opposite the smallest interior angle. Similarly, the longest
side is always opposite the largest interior angle. E/C DACALOS,
JAMES LOUIE

26. PRESENTATION OF FORMULAS AND
PROPERTIES Rectangle Properties
The fundamental properties of rectangles are:
* A rectangle is a quadrilateral
* The opposite sides are parallel and equal to each other
* Each interior angle is equal to 90 degrees
* The sum of all the interior angles is equal to 360 degrees
* The diagonals bisect each other
* Both the diagonals have the same length
* It has 4 sides, 4 angles, and 4 corners (vertices). 2E/C DACALOS,
JAMES LOUIE

27. A cargo ship is designing a new sail in the
shape of an isosceles triangle. The base of
the sail, which runs along the ship's deck, is
10 meters long. Each of the two equal sides
of the sail measures 13 meters. What is the
area, altitude, and perimeter of the sail? TRIANGLE PROBLEMS
a=13m
c=13m
b=9m E/C GENOBIA,
JOHN ROMUEL

28. SOLUTION Given: a = 13m
b = 10m
c = 13m Asked: h, P Formula: h = √a²-b²
Perimeter= a+b+c Solution:
Solve for h:
h=√a² - b²
=√13² - 5²
=√169 - 25
=√144
h= 12 m
Solve for P:
P= a+b+c
= 13+10+13
P= 36 mE/C GENOBIA,
JOHN ROMUEL

29. A ladder is leaning against a wall forming a
right triangle. The ladder is 15m long
(hypotenuse) and makes an angle of 60
with the ground. Find the height of the wall
the ladder reaches and the distance of the
ladder’s base from the wall.
o TRIANGLE PROBLEMSE/C ABAN,
CHRISTIAN LARRY

30. SOLUTION Given: hypotenuse (c) = 15m
A = 60
O

Asked: h, b
Formula: sin = A/c
cos = b/c Solution:
Solve for h:
a = c x sin A
= 15 x sin (60)
= 15 x 0.866
a = 12.99 E/C ABAN,
CHRISTIAN LARRY Solve for h:
b = c x cos A
= 15 x cos 60
o
= 15 x 0.5
b = 7.5m

31. A rectangle shaped tv has a length of 16 cm
and a width of 32 cm. Find the diagonal. RECTANGLE PROBLEMS
L= 16cm
W= 32 cmE/C MONSANTO,
JHON MIKE

32. SOLUTION Given: L = 16cm
W = 32cm

Asked: d
Formula: d = √L + W
2 2Solution:
Solve for d:
d = √L + W
2 2
= √16 + 32
2 2
= √1280
d = 35.77 cm E/C MONSANTO,
JHON MIKE

33. A rectangle shaped table has a length of 20
cm and a width of 30 cm. What is its area
and perimeter? TRIANGLE PROBLEMS
L= 20 cm
W = 30 cmE/C BUSTAMANTE,
STEPHEN

34. SOLUTION Given: L = 20cm
W = 30cm

Asked: A, P
Formula: A=bh
P=2(L+W)Solution:
Solve for A:
A = LW
= 20 (30)
A = 600 sq. cm Solution:
Solve for P:
P = 2 (L+W)
= 2 (20+30)
P = 100 cm E/C BUSTAMANTE,
STEPHEN

35. IMPORTANCETRIANGLE
Triangles and rectangles are essential geometric shapes that
play a vital role in many areas of everyday life, particularly in
engineering and construction. These shapes are found in
structures all over the world, from bridges and buildings to
roofs and towers. Their importance lies in their ability to
provide strength, balance, and stability. Triangles are known
for their structural efficiency, as they distribute weight
evenly and resist bending, making them ideal for supporting
heavy loads. Rectangles, on the other hand, are used to
create frames, walls, and organized layouts in both small-
scale and large-scale construction. These shapes are also
heavily studied in college-level courses, especially in
engineering, where they are used in calculations, design,
measurements, and structural planning.E/C SELLEZA, SHRIEP MAVERICK

36. IMPORTANCETRIANGLE
In the maritime industry, triangles and rectangles are just as
important. Shipbuilders use these shapes to design vessels
that are strong, stable, and efficient in water. Triangular
components are often found in the ship’s internal framework
to ensure balance and durability, while rectangles help form
the ship’s compartments and outer structure. Beyond
construction, these shapes are also used in maritime
navigation. Triangular angles are applied in tools and methods
used to calculate direction and distance, helping ships stay on
course. In conclusion, triangles and rectangles are more than
just simple shapes—they are fundamental tools that
support modern engineering, guide navigation, and form the
backbone of countless structures both on land and at sea.E/C SELLEZA, SHRIEP MAVERICK

37. APPLICATIONS OF TRIANGLES IN
MARINE ENGINEEING TRIANGLE NAVIGATION USING SPHERICAL TRIGONOMETRY
• Triangles are essential in calculating distances and courses on Earth’s
curved surface.
• Used in Great Circle and Rhumb Line navigation to plot efficient ship routes.
MARINE RAMP DESIGN
• Triangular principles help design sloped surfaces like ramps and staircases
on ships.
• Ensures proper angle and load distribution for safe movement between
decks.
STRUCTURAL ANALYSIS AND LOAD DISTRIBUTION
• Triangular trusses are used in ship hulls and offshore platforms for
stability.
• Triangles resist deformation, making them ideal for stress-bearing
structures.E/C TAMAYO, IAN

38. APPLICATIONS OF RECTANGLES IN
MARINE ENGINEEING TRIANGLE SHIP INTERIOR LAYOUT
• Rectangular shapes dominate cabins, decks, and bulkheads for efficient
space use.
• Simplifies modular construction and maximizes usable area.
MARINE ANTENNA DESIGN
• Rectangular microstrip patch antennas are used for wireless
communication at sea.
• Dimensions affect signal strength, gain, and directivity.
ENGINEERING CALCULATIONS
• Rectangles are used in calculating second moments of area for
structural design.
• Helps engineers assess strength and stability of ship components.E/C TAMAYO, IAN

39. REFERENCES https://byjus.com/maths/triangles/
https://geniebook.com/tuition/primary-6/maths/brief-history-geometrical-shapes
https://www.storyofmathematics.com/rectangle/#:~:text=1:%20Generic%20rectangle.-,Historical%20Significance,Ancient%20Greece
https://www.splashlearn.com/math-
vocabulary/geometry/rectangle#:~:text=A%20rectangle%20is%20a%20closed,width%20is%20the%20shorter%20side.
https://www.theseus.fi/bitstream/handle/10024/493217/TuomiMikko.pdf?sequence=2
https://www.irjmets.com/uploadedfiles/paper/issue_8_august_2023/44211/final/fin_irjmets1692949358.pdf
https://www.geeksforgeeks.org/maths/rea-life-applications-of-rectangles/
https://www.ripublication.com/ijoo20/ijoov14n1_06.pdf
https://usna.edu/NAOE/_files/documents/Courses/EN400/02.01%20Chapter%201.pdf