Lecture2: Solutions and Application of Right Triangle (Pythagorean Theorem)

SOLUTIONS AND APPLICATIONS OF RIGHT TRIANGLE

RIGHT TRIANGLE – is a triangle with exactly one (1) right angle.

Parts of a Right Triangle

1.       Hypotenuse – is the side opposite the 90^o angle

and it is also considered to be the longest side.

2.       Legs–these are the remaining sides of a

right triangle which may be shorter or longer.

3.       Angles

Pythagorean Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the square of its legs.
In symbols:

c2 = a2 + b2             or         a2 = c2– b2               or         b2 = c2 – a2

e.g.

1)      a = 8; b = 15; c = ?

c² = a² + b²

c² = 8² + 15²

c² = 64+225

c = sqrt 289

c = 17

2)      a = ?; b = 4; c = 5

a² = c²– b²

a² = 5²– 4²

a² =25– 16

a = sqrt 9

a = 3

3)      a = 12; b = ?; c = 37

b² = c² – a²

b² = 37² – 12²

b² = 1369 –144

b = sqrt 1225

b = 35

Sample Problems

      1)      A wall 12ft high cast a shadow 5ft long from the foot. Find the distance from the top of the wall to the tip of the shadow.

Illustration:

Given: a = 5ft; b = 12ft; c = ?

Solution:c² = a² + b²

                c² = (5ft)² + (12ft)²

               c² = 25ft² + 144ft²

              c = sqrt 169

              c = 13ft

2)      A ladder 9m long is placed 6m from the foot of a tree. Up to what height does the ladder reach the tree?

Illustration:

Given: a = 6m; b =?; c = 9m

Solution:b² = c²– b²

                b² = (9m)² – (6m)²

            b² = 81m² –36m²

            b = sqrt 45

b =  6.71m

Lecture1 : Overview

TRIGONOMETRY: Overview

Definition:

TRIGONOMETRY comes from three Greek words: “tri” meaning THREE; “gonia” meaning ANGLE; and “metrein” or “metron” meaning MEASUREMENT.

Generally, TRIGONOMETRY is a branch of Mathematics which deals with the study of triangles, its properties and measures.

Approaches in the study of Trigonometry

1.       Triangular Approach – deals with trigonometric function.

2.       Circular Approach – deals with circular function.

Angles and its Classifications

ANGLES – is a plane figure having two or more rays joining in a common endpoint.

Parts of an Angle

1.       Vertex – the common endpoint in an angle.

2.       Initial side – the ray of an angle wherein it starts to open.

3.       Terminal side – the ray of an angle wherein it stops to open.

Naming an Angle

1.       By Vertex

2.       By 3 (three) points

3.       By Greek letters: e.g. a; b; g; q; f; l; j

Unit measure of an Angle

1.       Degree measure / DMS

2.       Radian measure

Sign of an Angle

1.       Positive angle – an angle is said to be positive if it rotates in a counter-clockwise direction.

2.       Negative angle – an angle is said to be negative if it rotates in a clockwise direction.

Classification of Angles

1.       Zero angle – is an angle whose measure is exactly 0^o.

2.       Acute angle – is an angle whose measure is greater than 0^o but less than 90^o.

3.       Right angle – is an angle whose measure is exactly 90^o.

4.       Obtuse angle – is an angle whose measure is greater than 90^o but less than 180^o.

5.       Straight angle – is an angle whose measure is exactly 180^o.

6.       Reflex angle – is an angle whose measure is greater than 180^o but less than 360^o.

7.       Full angle/Angle of complete revolution – is an angle whose measure is exactly 360^o.

Complementary angles – are two angles whose sum is equal to 90^o.

Supplementary angles – are two angles whose sum is equal to 180^o.

Triangles and Its Classifications

TRIANGLE – is a closed plane figure or polygon with

three sides and three angles.

General Property of a Triangle

“The sum of all interior angles of any triangle is equal to 180^o.”

Classification of Triangles

A.      According to SIDES:

1.       Isosceles Triangle – is a triangle with two (2) equal sides.

2.       Equilateral Triangle – is a triangle with all three (3) equal sides.

3.       Scalene Triangle – is a triangle with no equal sides.

                                                                                                   

B.      According to ANGLES:

1.       Acute Triangle – is a triangle with all angles is acute.

2.       Obtuse Triangle – is a triangle with one obtuse angle.

3.       Right Triangle – is a triangle with exactly one right angle.

4.       Equiangular Triangle – is a triangle with all equal angles.