GEOMETRYFourth Grading

CONTENT STANDARD

The learners demonstrates understanding of

the basic concepts of trigonometry.

PERFORMANCE STANDARD

The learner is able to apply the concepts of

trigonometric ratios to formulate and solve

real-life problems with precision and

accuracy.

COMPETENCY

The learner illustrates law of sines and

cosines. (M9GE-IVf-g-1)

PRE-REQUISITE SKILLS

• Illustrates the six trigonometric ratios: sine,

cosine, tangent, secant, cosecant, and

cotangent. (M9GE-IVa-1)

• Finds the trigonometric ratios of special

angles. (M9GE-IVb-c-1)

• Illustrates the Pythagorean Theorem.

ACTIVITY 1

1. What does h

represents in the

∆ABC and what are

its properties?

• h is one of the altitudes ∆ABC.

• It is perpendicular to AB.

• It divides the ∆ABC into two triangles.

ACTIVITY 1

2. In ∆ADC, write the

equation that relates

b2 to h2 and x2. Justify

your answer.

By Pythagorean Theorem:

b2 = x2 + h2

ACTIVITY 1

3. Write the equation

that relates x to b and

∠A. Justify your

answer.

By trigonometric ratio:

x = b cos A

ACTIVITY 1

4. Write the equation

that relates a to h and

c – x in expanded

form.

a2 = h2 + (c – x)2

a2 = h2 + c2 – 2cx + x2

ACTIVITY 1

5. Rearrange your

equation such that

the term x2 is next to

the term h2 or vice

versa.

a2 = h2 + c2 – 2cx + x2

a2 = h2 + x2 + c2 – 2cx

ACTIVITY 16. Underline the part

in your equation in

number 5 which is

similar to your answer

in number 2.

Substitute this

expression into the

expression you wrote

in number 2 and write

it below.

a2 = h2 + x2 + c2 – 2cx

a2 = b2 + c2 – 2cx

ACTIVITY 1

7. Substitute x with

your answer in

number 3 and write it

below.

a2 = b2 + c2 – 2cx

a2 = b2 + c2 – 2cb cos A

ACTIVITY 2

1. Draw an altitude h

from B. Write the

equation of h in terms

of ∠A and c. Also the

equation of h in terms

of ∠C and a.A

B

C

ac

b

h = c sin A h = a sin C

h

ACTIVITY 2

2. Are the two

equations the same?

Justify your answer

and write the equation

that represents the

relationship.A

B

C

ac

b

Yes, since both equations are equal to h.

c sin A = a sin C

h

ACTIVITY 23. Write your equation

in number 2 in a form

of proportion (fractional

form). What property of

proportion that can

support the process of

rewriting the equation?A

B

C

ac

b

By converse of cross-multiplication property,

h

c sin A = a sin Cc

sin C=

a

sin A

ACTIVITY 24. Draw an altitude i

from A. Write the

equation relating i to

∠C and b. Also, write

the equation relating i

to ∠B and c in

fractional form.A

B

C

ac

b

𝑖 =𝑐

sin C𝑖 =

𝑏

sin B

h

i

ACTIVITY 25. Write a new

equation relating the

two equations together.

A

B

C

ac

b

𝑐

sin C=

𝑏

sin B

h

i

ACTIVITY 26. Using the Transitive

Property of Equality,

write an equation

relating the equations

in #3 and #5.

A

B

C

ac

b

𝑎

sin A=

𝑏

sin B=

𝑐

sin C

h

i