የምዕራፍ ማስታወሻዎችና የመልመጃ ጥያቄዎች

የ9ኛ ክፍል ሒሳብ ምዕራፍ 5፦ የትሪጎኖሜትሪ መግቢያ

በዚህ ገጽ ያለው ሁሉ ለ9ኛ ክፍል የሒሳብ መፅሐፍ ምዕራፍ 5 ነው፦ ምዕራፉ የሚሸፍነው፣ አጭር ማስታወሻዎቹ፣ የመልመጃ ጥያቄዎች ከመልሶቻቸው ጋር፣ እና ቪዲዮ ባለበት የትምህርቱ ቪዲዮ።

4
ካርዶች: 10
የምዕራፍ 5 የመልመጃ ጥያቄዎች: 162–180
የምዕራፍ 5 የመፅሐፍ ገጾች

ምዕራፍ 5 የሚሸፍነው

1.Revision on Right-angled Triangles
2.The Trigonometric Ratios

የምዕራፍ 5 አጭር ማስታወሻዎች

በ9ኛ ክፍል የሒሳብ ገጽ ላይ በዚህ ምዕራፍ ሥር የተመዘገቡት ካርዶች። አንድ በአንድ ይክፈቱ።

Confusing Opposite and Adjacent Sidesየተለመደ ስህተት

መቼ እንደሚጠቀሙበት

ለአንድ አጣዳፊ ማዕዘን የ sine, cosine ወይም tangent ንፅፅር ከማስላትዎ በፊት ይህንን ያረጋግጡ።

ይጠንቀቁ

hypotenuse ሁልጊዜ ከ90 ዲግሪ ማዕዘን ትይዩ ሆኖ የማይለወጥ ሲሆን፣ opposite እና adjacent ግን ከአንዱ አጣዳፊ ማዕዘን ወደ ሌላው ሲቀየሩ ቦታ ይለዋወጣሉ።

ከGrade 9 Mathematics፣ ገጽ 162-180 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል

Pythagoras Theoremቀመር

b^2 + p^2 = h^2

b^2 + p^2 = h^2

$h$: ሰያፍ ጎን $m$
$b$: መሰረት $m$
$p$: ቋሚ ጎን $m$

መቼ እንደሚጠቀሙበት

የቀኝ ማዕዘን ሶስት ማዕዘን ያልታወቀ ጎን ርዝመት ለማግኘት ወይም ሶስት ማዕዘኑ ቀኝ ማዕዘን መሆኑን ለማረጋገጥ ይጠቀሙበት።

ይጠንቀቁ

hypotenuse h ሁልጊዜ ከ90 ዲግሪ ማዕዘን ትይዩ የሚገኝ ረጅሙ ጎን መሆን አለበት። ጎኖችን በh ቦታ በስህተት አያስቀምጡ።

ከGrade 9 Mathematics፣ ገጽ 162-180 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል

Trigonometric Ratiosቀመር

\sin A = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos A = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan A = \frac{\text{opposite}}{\text{adjacent}}

\sin A = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos A = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan A = \frac{\text{opposite}}{\text{adjacent}}

$A$: ሹል አንግል $^\circ$
$\text{opposite}$: ከአንግል A ትይዩ የሆነ ጎን $m$
$\text{adjacent}$: ለአንግል A አጎራባች የሆነ ጎን $m$
$\text{hypotenuse}$: ሰያፍ ጎን $m$

መቼ እንደሚጠቀሙበት

በቀኝ ማዕዘን ሶስት ማዕዘን ውስጥ አጣዳፊ ማዕዘንን ከጎኖች ርዝመት ጋር ለማዛመድ ይጠቀሙባቸው።

ይጠንቀቁ

opposite እና adjacent የሚወሰኑት በተመረጠው ማዕዘን ላይ በመመስረት ነው። ለማዕዘን A ትክክለኛውን ጎን መለየትዎን ያረጋግጡ።

ከGrade 9 Mathematics፣ ገጽ 162-180 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል

Trigonometric Values of Special Anglesየማጣቀሻ ሠንጠረዥA

Ratio	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin A$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos A$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\tan A$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

መቼ እንደሚጠቀሙበት

ሠንጠረዥ ሳያስፈልግ የልዩ ማዕዘናትን ትክክለኛ የ trigonometric ratio ዋጋዎች ለማግኘት ይጠቀሙበት።

ይጠንቀቁ

የ tan 90 ዲግሪ ዋጋ ለዜሮ ማካፈል ስለሚከሰት አይተረጎምም (undefined)።

ከGrade 9 Mathematics፣ ገጽ 162-180 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል

የምዕራፍ 5 የመልመጃ ጥያቄዎች

በዚህ ምዕራፍ ላይ የተጻፉ 10 የምርጫ ጥያቄዎች፣ እያንዳንዳቸው መልሱና ምክንያቱ አላቸው። መልሱን ከመክፈትዎ በፊት ይሞክሩ። ጥያቄዎቹ በእንግሊዝኛ ናቸው።

ጥያቄ 1

In a right-angled triangle, what is the special name given to the side opposite to the 90 degree angle?

A.Base
B.Perpendicular
C.Hypotenuse
D.Adjacent

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ C Hypotenuse

ምክንያቱ

By definition, the side opposite to the right angle (90 degrees) in a right-angled triangle is called the hypotenuse, and it is always the longest side.

ጥያቄ 2

What is the maximum degree measure of an interior angle in a right-angled triangle?

A.45 degrees
B.60 degrees
C.90 degrees
D.180 degrees

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ C 90 degrees

ምክንያቱ

A right-angled triangle has one angle equal to 90 degrees, and the sum of the other two acute angles is 90 degrees. Therefore, the maximum interior angle measure is 90 degrees.

ጥያቄ 3

The word trigonometry is derived from two Greek words: trigonon and metron. What does trigonon mean?

A.Measure
B.Triangle
C.Angle
D.Circle

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ B Triangle

ምክንያቱ

According to the textbook, trigonon means triangle and metron means measure.

ጥያቄ 4

For an acute angle A in a right-angled triangle, how is the tangent ratio (tan A) defined?

A.Opposite side divided by adjacent side
B.Adjacent side divided by hypotenuse
C.Opposite side divided by hypotenuse
D.Hypotenuse divided by adjacent side

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ A Opposite side divided by adjacent side

ምክንያቱ

By definition, tan A = (side opposite to angle A) / (side adjacent to angle A).

ጥያቄ 5

Which of the following sets of side lengths can form a right-angled triangle?

A.5, 6, 2
B.4, 6, 8
C.1, 2, √3
D.2, 3, 4

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ C 1, 2, √3

ምክንያቱ

Using the converse of Pythagoras theorem: for side lengths 1, √3, and 2 (where 2 is the longest side), 1² + (√3)² = 1 + 3 = 4, and 2² = 4. Since LHS = RHS, these sides form a right-angled triangle.

ጥያቄ 6

An equilateral triangle has sides of length 6 units. What is the height of this triangle?

A.3√3 units
B.3√2 units
C.2√5 units
D.3 units

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ A 3√3 units

ምክንያቱ

Drawing a perpendicular height splits the base of 6 into two segments of 3 units each. By Pythagoras theorem, h² + 3² = 6², which gives h² + 9 = 36, so h² = 27, and h = 3√3 units.

ጥያቄ 7

What is the exact value of tan 60°?

A.1 / √3
B.1
C.√3
D.√3 / 2

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ C √3

ምክንያቱ

In a standard 30°-60°-90° right-angled triangle, the side opposite to 60° is √3 and the adjacent side is 1. Thus, tan 60° = √3 / 1 = √3.

ጥያቄ 8

As the acute angle A increases from 0° to 90°, what happens to the values of sin A and cos A?

A.sin A increases from 0 to 1, while cos A decreases from 1 to 0
B.sin A decreases from 1 to 0, while cos A increases from 0 to 1
C.Both sin A and cos A increase from 0 to 1
D.Both sin A and cos A decrease from 1 to 0

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ A sin A increases from 0 to 1, while cos A decreases from 1 to 0

ምክንያቱ

As noted in the textbook remark, as ∠A increases from 0° to 90°, sin A increases from 0 to 1 while cos A decreases from 1 to 0.

ጥያቄ 9

In a right-angled triangle, if an acute angle A has sin A = 3/5, what is the value of tan A?

A.4/5
B.3/4
C.4/3
D.5/3

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ B 3/4

ምክንያቱ

Since sin A = opposite / hypotenuse = 3/5, let opposite = 3 and hypotenuse = 5. By Pythagoras theorem, adjacent = √(5² - 3²) = √(25 - 9) = √16 = 4. Therefore, tan A = opposite / adjacent = 3/4.

ጥያቄ 10

If A is an acute angle in a right-angled triangle and tan A = 1 / √2, what is the value of cos A?

A.1 / √3
B.√2 / √3
C.√3 / √2
D.1 / 2

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ B √2 / √3

ምክንያቱ

With tan A = opposite / adjacent = 1 / √2, opposite = 1 and adjacent = √2. Hypotenuse = √(1² + (√2)²) = √(1 + 2) = √3. Therefore, cos A = adjacent / hypotenuse = √2 / √3.

መተግበሪያው በዚህ ምዕራፍ ላይ 12 ጥያቄዎች አሉት፤ እየሠሩ ያርምዎታል። እዚህ ያሉት የመጀመሪያዎቹ 10 ናቸው።