MATH MODELS IN
REAL LIFE
By: Nathaniel Anabwani
INSPIRED BY
Pythagorean Theorem
GEOMETRY Trigonometric Ratios
TOPICS Triangle Similarity
Elevation and Depression
Congruent triangles
PYTHAGOREAN THEOREM
The Pythagorean theorem states that in a right
triangle, the square of the length of the hypotenuse
(the side opposite the right angle) is equal to the sum
of the squares of the lengths of the other two sides.
Mathematically, this can be expressed as ( a^2 + b^2
= c^2 ), where ( a ) and ( b ) are the lengths of the two
shorter sides (legs) of the triangle, and ( c ) is the
length of the hypotenuse. This theorem is important
in geometry and trigonometry, allowing us to
calculate unknown side lengths or angles in right
triangles.
TRIGONOMETRIC RATIOS
Trigonometric ratios is a topic in trigonometry that compares the
angles of a right triangle to the lengths of its sides. The three main
trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). Sine is
defined as the ratio of the length of the side opposite an angle to the
length of the hypotenuse, cosine is the ratio of the length of the
adjacent side to the hypotenuse, and tangent is the ratio of the
opposite side to the adjacent side. These ratios are used extensively in
solving problems involving triangles, such as finding missing side lengths
or angles. Additionally, the reciprocal trigonometric ratios, cosecant
(csc), secant (sec), and cotangent (cot), are the inverses of sine, cosine,
and tangent, and represent the ratios of the lengths of the
hypotenuse to the opposite side, hypotenuse to the adjacent side.
TRIANGLE SIMILARITIES
Triangle similarity is the relationship between
two triangles that have corresponding angles that are
equal. This is shown by the AA (angle-angle), SAS (side-
angle-side), and SSS (side-side-side) similarity criteria. In
AA similarity, if two angles in one triangle are congruent
to two angles in another triangle, then the triangles are
similar. SAS similarity states that if two sides of one
triangle are proportional to two sides of another
triangle and the included angle is congruent, then the
triangles are similar. Similarly, SSS similarity means that
if the lengths of the corresponding sides of two
triangles are proportional, then the triangles are similar.
This similarity is essential in geometry and are used to
prove various properties and solve geometric problems
involving similar triangles.
ANGLE ELEVATION/DEPRESSION
In geometry, angle elevation and depression refer
to the orientation of angles in relation to the
horizontal plane. Angle elevation happens when an
angle is measured above the horizontal plane, like
looking up at an object or measuring the angle of
elevation of a hill. On the other hand, angle
depression also occurs when an angle is measured
below the horizontal plane, like looking down from
a high point or measuring the angle of depression
of a building from an observer's point of view.
These concepts are important in various
geometric calculations, including determining
heights, distances, and angular relationships in
trigonometry.
CONGRUENT
TRIANGLES
Congruent triangles is definitely one of the most confusing topics we have
covered as well. More specifically, the way it was explained to me at least,
the acronyms ASA, SSA, SSS, SAS, RHS were super mind boggling. I say this
because the mini congruency lines could easily confuse you due to multiple
triangles being connected together (or a rectangle being cut into even
shapes diagonally). Congruent triangles are defined as three corresponding
sides that are equal and that all the three corresponding angles are equal in
measure. These triangles can also be slid, rotated, flipped, turned to look
identical.
TODAY, YOU
Pythagorean Theorem
LEARNED Trigonometric Ratios
TO... Triangle Similarites
Angle Elevation/Depression
Congruent Triangles
