Understanding Trigonometry in Taglish: Lesson 5 - Triangles

Lesson 5 - Triangles

Bago natin suungin ang aralin tungkol sa triangle o trianggulo na siyang pinakabasehan ng Trigonometry, alalahanin nating muli ang ating mga napag-aralan na:

• An angle is formed by two rays with a common endpoint. The two rays are called the sides of the angle and the common endpoint is called the vertex.

• Angles are usually named using three capital letters of the English alphabet. But they can also be named using numbers or letters of the Greek alphabet.

• A protractor is an instrument used to measure an angle. The units of measurement for angles are degrees (°) or radians (r).

• Angles are classified according to their measurements.

• Acute angle — an angle whose measurement is less than 90°.

• Right angle — an angle whose measurement is exactly 90°.

• Obtuse angle — an angle whose measurement is more than 90°

but less than 180°.

• Straight angle — an angle whose measurement is exactly 180°.

• Reflex angle — an angle whose measurement is more than 180°

but less than 360°.

• If the sum of the measurements of two angles is 90°, they are complementary angles.

• To find the measurement of the complement of a given angle, subtract the measurement of the given angle from 90°.

• If the sum of the measurements of two angles is 180°, they are supplementary angles.

• To find the measurement of the supplement of a given angle, subtract the measurement of the given angle from 180°.

After learning all about angles in the previous lesson, you are now ready to tackle more complicated figures such as triangles. This lesson will focus on the different kinds of triangles. The knowledge you will gain from this lesson will hopefully help you in solving real-life problems. 

Matapos mong malaman ang lahat tungkol sa mga anggulo sa nakaraang aralin, handa ka na ngayon upang harapin ang mas kumplikadong mga pigura tulad ng mga triangles. Ang aralin na ito ay magtutuon sa iba't ibang mga uri ng mga triangles. Ang kaalamang makukuha mo mula sa araling ito ay sana makatulong sa iyo sa paglutas ng mga problema sa totoong buhay.

Ano ang triangle?

A triangle is a polygon with three sides and three angles.

Ang trianggulo ay isang polygon na may tatlong gilid at tatlong anggulo.

Masdan ang triangle sa itaas. Ang simbolo para sa isang trianggulo ay D. Tatawagin nating  D QRP ang pigura sa itaas. Ang sides o mga gilid ng D QRP ay QR (RQ), QP (PQ), at RP (PR). Ang tatlong angles naman ay ÐQ,  ÐR, at ÐP.

What are the different kinds of triangles?

A. Triangles are classified according to the lengths of their sides.

Ang mga trianggulo ay kinaklase ayon sa haba ng mga sides/gilid nito.

1. A scalene triangle is a triangle with no congruent side. Congruent means having the same or equal measure.

In the above figure, PQ ≠ QR ≠ PR, so ΔPQR is a scalene triangle.

2. An isosceles triangle is a triangle with two congruent sides.

3. An equilateral triangle is a triangle with three congruent sides.

B. Triangles are also classified according to the kinds of angles that make them up.

1. A triangle with a right angle is called a right triangle. A right angle measures 90 degrees.

2. A triangle that is made up of three acute angles is called an acute triangle. An acute angle measures more than zero degrees but less than 90 degrees.

3. A triangle with an obtuse angle is called an obtuse triangle. An obtuse angle measures more than 90 degrees but less than 180 degrees.

In summary, triangles are classified according to the lengths of their sides and the angles that they make up. See the illustration below from  https://www.cuemath.com to understand and remember the concept:

Pagsasanay

        Upang matasa kung nauunawaan ang leksyon, sagutin ang mga sumusunod na katanungan:

1. Magkakasukat ang tatlong gilid o sides ng isang trianggulo. Ang triangle na ito ay tinatawag na __________ triangle.

A. right 

B. equilateral

C. isosceles

D. acute

2. Isa sa mga angles ng triangle na ito ay may sukat na 91 degrees. Ang triangle na ito ay isang ________ triangle.

A. right

B. acute

C. obtuse

D. isosceles

3. Ang isosceles triangle ay may magkakaparehong sukat ng _______ gilid o sides.

A. isa

B. dalawa

C. tatlo

D. apat

4. May tatlong acute angles ang triangle na ito na hindi magkakapareho ang sukat. ________ triangle ang tawag dito.

A. acute

B. scalene

C. obtuse

D. right

5. Ang mga sides o gilid ng triangle na ito ay iba-iba ang mga sukat. Ang tawag sa triangle na ito ay ________ triangle.

A. equilateral

B. isosceles

C. reflex

D. scalene

6. Ang sukat ng  mga angles ng isang equilateral triangle ay _________.

A. iba-iba

B. 30 degrees

C. 45 degrees

D. 60 degrees

7. Ang supplementary angle ng ÐY = 30 degrees ay _______ degrees.

A. 60

B. 90

C. 150

D. 330

8. Alin sa mga sumusunod na pares ang complementary angles?

A. Ð2  = 30^o    |     ÐA = 30^o

B. Ð3  = 35^o    |     ÐB = 35^o

C. Ð4   = 45^o   |     ÐC = 45^o

D. Ð5   =  90^o  |     ÐM = 90^o

9. Ang triangle na ito ay may dalawang gilid na magkakapareho ang sukat. _______ triangle ang tawag dito.

A. isosceles

B. equilateral

C. complementary

D. right

10. Ilang right angle mayroon ang isang right triangle?

A. wala

B. isa

C. dalawa

D. tatlo

Mga sagot sa pagsasanay

Understanding Trigonometry in Taglish: Lesson 4 - SOLVING REAL-LIFE PROBLEMS USING THE PYTHAGOREAN THEOREM

 LESSON 4 – SOLVING REAL-LIFE PROBLEMS USING THE PYTHAGOREAN THEOREM

Matapos nating mapag-aralan ang mga uri ng linya, salikop o anggulo, complementary at supplementary angle gayundin ang katuturan ng Pythagorean Theorem, sa leksyon 4 ay susubukin naman nating mabigyang solusyon ang mga halimbawa ng mga problema sa tunay na buhay gamit ang bisa ng Pythagorean Theorem ni Pythagoras.

Dapat lamang nating tandaan na ang Pythagorean Theorem ay ginagamit lamang kung may mabubuo tayong right angle o right triangle.

STEPS in solving problems involving RIGHT TRIANGLES:

Nasa ibaba ang mga hakbang upang masolusyonan ang isang Math problem na kinassangkutan ng Right Triangle:

STEP 1 Identify the hypotenuse and the legs of the triangle. (Tukuyin ang hypotenuse at mga legs ng triangle.)

STEP 2 Use the formula derived from the Pythagorean theorem. (Gamitin ang formula ng Pythagorean theorem: c2 = a2 + b2)

STEP 3 Substitute the given values to the variables in the formula and solve for the unknown. (Ilagay ang mga values na ibinigay sa problema at hanapin ang unknown.)

STEP 4 Simplify to get the final answer. (Gawing simple ang sagot)

[Problem 1 to 4 from ALS Junior High School Module Trigonometric Functions 1]

Problem 1

Jake was performing his favorite billiard trick shot. He hit the cue ball and it went to one corner of the table, rolled to the other corner, hit the exact center of the back cushion and hit the eighth ball causing it to drop into one of the holes before returning to its original position. If the dimensions of the table are 6 ft. by 8 ft., how far did the cue ball travel?

Problem 2

Alice, a landscape architect, designed a flowerbed for a very important client. The flowerbed will take the shape of a right triangle with legs equal to 12 ft. and 9 ft. She wants to place some decorative stones along the hypotenuse of the flowerbed 1 ft. apart. How many stones does she need?

Problem 3

Two cars start traveling perpendicularly away from each other from the same place. If one travels 6 km and the other 8 km, how far apart will they be from each other using their bumpers as points of reference? 

Problem 4

A pendulum travelled 30 cm away from a center line. The center line is 40 cm long. What is the distance between the pendulum and one end of the center line?

Problem 5

A ladder 13 m long is placed on the ground in such a way that it touches the top of a vertical wall 12 m high. Find the distance of the foot of the ladder from the bottom of the wall. 

(Source: https://www.math-only-math.com)

TANDAAN

1. Magagamit lamang ang Pythagorean Theorem sa  mga right angles o right triangles.

2. Hindi lang sukat ng hypotenuse ang mahahanap sa Pythagorean Theorem kundi maging ang iba pang sukat/haba ng mga sides/legs (opposite and adjacent sides) kung ang 2 sukat ay ibinigay.

3. Mag-drawing kung hindi ma-imagine ang larawang isinasaad ng problem.

4. Tandaan ang square root ng ilang bilang tulad ng:

1 = square root ng 1 dahil 1 x 1 or 1² = 1

2 = square root ng 4 dahil 2 x 2 or 2² = 4

3 = square root ng 9 dahil 3 x 3 or 3² = 9

4 = square root ng 16 dahil 4 x 4 or 4² = 16

5 = square root ng 25 dahil 5 x 5 or 5² = 25

6 = square root ng 36 dahil 6 x 6 or 6² = 36

        7 = square root ng 49 dahil 7 x 7 or 7² = 49

8 = square root ng 64 dahil 8 x 8 or 8² = 64

9 = square root ng 81 dahil 9 x 9 or 9² = 81

10 = square root ng 100 dahil 10 x 10 or 10² = 100

11 = square root ng 121 dahil 11 x 11 or 11² = 121

12 = square root ng 144 dahil 12 x 12 or 12² = 144

13 = square root ng 169 dahil 13 x 13 or 13² = 169

14 = square root ng 196 dahil 14 x 14 or 14² = 196

15 = square root ng 225 dahil 15 x 15 or 15² = 225

Alalahanin na ang 

A = square root ng Y dahil A x A (A times A) or A² = Y

A square root of a number (Y) is one of the factors (A)  of Y such that when you multiply it TWO (2) times (A x A) or you square it (A²), you will get that number Y.