የምዕራፍ ማስታወሻዎችና የመልመጃ ጥያቄዎች

የ9ኛ ክፍል ሒሳብ ምዕራፍ 7፦ ገጥምነት እና ተመሳስሎ

በዚህ ገጽ ያለው ሁሉ ለ9ኛ ክፍል የሒሳብ መፅሐፍ ምዕራፍ 7 ነው፦ ምዕራፉ የሚሸፍነው፣ አጭር ማስታወሻዎቹ፣ የመልመጃ ጥያቄዎች ከመልሶቻቸው ጋር፣ እና ቪዲዮ ባለበት የትምህርቱ ቪዲዮ።

4
ካርዶች: 10
የምዕራፍ 7 የመልመጃ ጥያቄዎች: 209–246
የምዕራፍ 7 የመፅሐፍ ገጾች

ምዕራፍ 7 የሚሸፍነው

1.Revision on Congruency of Triangles
2.Definition of Similar Figures
3.Theorems on Similar Plane Figures
4.Ratio of Perimeters of Similar Plane Figures
5.Ratio of Areas of Similar Plane Figures
6.Construction of Similar Plane Figures
7.Applications on Similarities

የምዕራፍ 7 አጭር ማስታወሻዎች

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

Linear Scale Factor እና የ Area ለውጥ ስህተትየተለመደ ስህተት

መቼ እንደሚጠቀሙበት

የጎን ርዝመት ሲቀየር የ Perimeter ወይም የ Area ለውጥን የሚጠይቁ ጥያቄዎች ሲያጋጥሙ ይህንን ያረጋግጡ።

ይጠንቀቁ

የጎን ርዝመትን በ 2 ማባዛት Perimeter እጥፍ ሲያደርግ Area ግን በ 2^2 = 4 እጥፍ ያሳድጋል።

ከGrade 9 Mathematics፣ ገጽ 209-246 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል

የተመሳሳይ ምስሎች የ Area ንፅፅርቀመር

\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2 = k^2

\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2 = k^2

$A_1$: የመጀመሪያው ተመሳሳይ ምስል Area $\text{m}^2$
$A_2$: የሁለተኛው ተመሳሳይ ምስል Area $\text{m}^2$
$s_1$: የመጀመሪያው ምስል side length $\text{m}$
$s_2$: የሁለተኛው ምስል ተጓዳኝ side length $\text{m}$
$k$: የስኬል ፋክተር $1$

መቼ እንደሚጠቀሙበት

የጎን ርዝመት ወይም የ scale factor ተሰጥቶ የተመሳሳይ ምስል Area ለማስላት ይጠቀሙ።

ይጠንቀቁ

የ scale factor ዋጋን ስኩዌር ማድረግ እንዳይረሱ። የ Area ንፅፅር k^2 እንጂ k አይደለም።

ከGrade 9 Mathematics፣ ገጽ 209-246 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል

የተመሳሳይ ምስሎች የ Perimeter ንፅፅርቀመር

\frac{P_1}{P_2} = \frac{s_1}{s_2} = k

\frac{P_1}{P_2} = \frac{s_1}{s_2} = k

$P_1$: የመጀመሪያው ተመሳሳይ ምስል Perimeter $\text{m}$
$P_2$: የሁለተኛው ተመሳሳይ ምስል Perimeter $\text{m}$
$s_1$: የመጀመሪያው ምስል side length $\text{m}$
$s_2$: የሁለተኛው ምስል ተጓዳኝ side length $\text{m}$
$k$: የስኬል ፋክተር $1$

መቼ እንደሚጠቀሙበት

ሁለት ጂኦሜትሪያዊ ምስሎች ተመሳሳይ ሲሆኑ ያልታወቀ Perimeter ወይም የጎን ርዝመት ለማግኘት ይጠቀሙ።

ይጠንቀቁ

ንፅፅሩን ሲያዘጋጁ ተጓዳኝ ጎኖችን በትክክለኛ ቅደም ተከተል ያዛምዱ።

ከGrade 9 Mathematics፣ ገጽ 209-246 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል

የትሪያንግል ተመሳሳይነት መስፈርቶችየማጣቀሻ ሠንጠረዥAA · SSS · SAS

መስፈርት	የተሰጠ እኩልነት ወይም ንፅፅር	መደምደሚያ
AA	$\angle A = \angle D \text{ and } \angle B = \angle E$	$\triangle ABC \sim \triangle DEF$
SSS	$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$	$\triangle ABC \sim \triangle DEF$
SAS	$\frac{AB}{DE} = \frac{AC}{DF} \text{ and } \angle A = \angle D$	$\triangle ABC \sim \triangle DEF$

መቼ እንደሚጠቀሙበት

ሁለት ትሪያንግሎች ተመሳሳይ መሆናቸውን በአነስተኛ የአንግል ወይም የጎን መረጃዎች ለማረጋገጥ ይህንን ይጠቀሙ።

ይጠንቀቁ

በ SAS ተመሳሳይነት ወቅት እኩል የሆነው አንግል በሁለቱ ተመጣጣኝ ጎኖች መካከል የሚገኝ መሆን አለበት።

ከGrade 9 Mathematics፣ ገጽ 209-246 የተዘጋጀ፤ ከመውጣቱ በፊት ሁለት ጊዜ ተረጋግጧል

የምዕራፍ 7 የመልመጃ ጥያቄዎች

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

ጥያቄ 1

According to the AA (Angle-Angle) similarity theorem, two triangles are similar if:

A.Two angles of one triangle are congruent to two angles of the other triangle
B.All three pairs of corresponding sides are equal in length
C.The two triangles have exactly equal perimeters
D.The two triangles have equal areas

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ A Two angles of one triangle are congruent to two angles of the other triangle

ምክንያቱ

The AA similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar.

ጥያቄ 2

The ratio of the perimeters of two similar triangles is 3:5. What is the ratio of their corresponding sides?

A.9:25
B.3:5
C.6:15
D.1:2

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ B 3:5

ምክንያቱ

For any two similar plane figures, the ratio of their perimeters is equal to the ratio of their corresponding side lengths, which is 3:5.

ጥያቄ 3

A giraffe is 6 m tall and casts a shadow of 4 m. At the same time, a smaller giraffe casts a shadow of 2 m. How tall is the smaller giraffe?

A.2 m
B.3 m
C.4 m
D.5 m

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ B 3 m

ምክንያቱ

Using the proportionality of similar triangles: Height / Shadow = 6 / 4 = x / 2. Solving for x gives x = (6 * 2) / 4 = 3 m.

ጥያቄ 4

If the ratio of the side lengths of two similar rectangles is 3:1, what is the ratio of their areas?

A.3:1
B.6:1
C.9:1
D.12:1

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ C 9:1

ምክንያቱ

The ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides: 3² : 1² = 9:1.

ጥያቄ 5

Given that ΔABC ~ ΔDEF, with AB = 9 cm, BC = 15 cm, and DE = 6 cm. What is the length of side EF?

A.8 cm
B.10 cm
C.12 cm
D.18 cm

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ B 10 cm

ምክንያቱ

Because the triangles are similar, the ratios of corresponding sides are equal: AB / DE = BC / EF, which means 9 / 6 = 15 / EF. Therefore, EF = (15 * 6) / 9 = 10 cm.

ጥያቄ 6

Two similar triangles have corresponding sides of length 3 cm and 9 cm. If the area of the smaller triangle is 12 cm², what is the area of the larger triangle?

A.36 cm2
B.72 cm2
C.108 cm2
D.144 cm2

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ C 108 cm2

ምክንያቱ

The scale factor of side lengths is 9 / 3 = 3. The ratio of their areas is 3² = 9. Thus, the area of the larger triangle is 12 * 9 = 108 cm².

ጥያቄ 7

In Hawassa, a flagpole casts a shadow 9 m long. At the same moment, a student who is 1.8 m tall standing nearby casts a shadow 4 m long. How tall is the flagpole?

A.3.60 m
B.4.05 m
C.4.50 m
D.5.20 m

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ B 4.05 m

ምክንያቱ

Set up the proportion: Height / Shadow = 1.8 / 4 = h / 9. Solving for height gives h = (1.8 * 9) / 4 = 16.2 / 4 = 4.05 m.

ጥያቄ 8

Two similar triangles have areas of 144 cm² and 169 cm². If a side of the larger triangle is 26 cm, what is the length of the corresponding side of the smaller triangle?

A.20 cm
B.22 cm
C.24 cm
D.25 cm

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ C 24 cm

ምክንያቱ

The ratio of corresponding sides is the square root of the ratio of areas: √(144 / 169) = 12 / 13. The side of the smaller triangle is 26 * (12 / 13) = 24 cm.

ጥያቄ 9

Abdi is 1.8 meters tall and casts a shadow of 2.4 meters. His younger brother casts a shadow of 2 meters at the same time. How tall is Abdi's brother?

A.1.2 meters
B.1.4 meters
C.1.5 meters
D.1.6 meters

መልሱንና ምክንያቱን ያሳዩ

መልስ፦ C 1.5 meters

ምክንያቱ

By similarity of triangles formed with the ground: Height / Shadow = 1.8 / 2.4 = h / 2. Solving for h gives h = (1.8 * 2) / 2.4 = 3.6 / 2.4 = 1.5 meters.

ጥያቄ 10

A photograph measuring 4 cm wide and 5 cm long is enlarged to make a wall mural in a hall in Addis Ababa. If the mural is 120 cm wide, what is the length of the mural?