252.

AB = 32 սմ
CO = 12 սմ
OD = 16 սմ

AO = x
OB = 32 – x
x(32 – x) = 12 * 16
32 – x2 = 196
-x² + 32 – 196 = 0
x² – 32 + 196 = 0

a = 1
b = -32
c = 196

D = b² – 4ac = (-32)² – 4 * 196 = 1024 – 784 = 240

x = ^{-b ± √D}/_2a = ^{32 ± √240}/₂ = ^{32 ± 4√15}/₂ = 16 ± 2√15
x₁ = 16 + 2√15 
x₂ = 16 – 2√15

AO = 16 + 2√15 
OB = 16 – 2√15 

Պատ․ AO = 16 + 2√15, OB = 16 – 2√15 

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253-ա.

ա) 4 սմ և 5 սմ

AD = 4 սմ, CD = 5 սմ
AC = 4 + 5 = 9 սմ

AB² = AD * AC
AB² = 4 * 9 = 36
AB = √36 = 6 սմ

Պատ․ AB = 6 սմ

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254.

AB = 20 սմ, AC = 50 սմ

AB² = AD * AC
20² = AD * 50
400 = AD * 50
AD = 400/50 = 8 սմ

CD = AC – AD = 50 – 8 = 42 սմ
r = CD/2 = 42/2 = 21 սմ

Պատ․ r = 21 սմ

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258.

AB = x
AD = x – 5
CD = x + 5
AB = ?

AB² = AC * AD
AC = AD + CD = x – 5 + x + 5 = 2x
x² = 2x * (x – 5)
x² = 2x² – 10x
x² – 10x = 0
x(x – 10) = 0
x ≠ 0
x – 10 = 0
x = 10 սմ

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259.

AC = 15 սմ
AE = 25 սմ
AD = x
AB = AD + 2 = x + 2

AC * AB = AD * AE
15(x + 2) = 25x
15x + 30 = 25x
30 = 10x
x = 3 սմ

AB = AD + 2 = 3 + 2 = 5 սմ

Պատ․ AB = 5 սմ, AD = 3 սմ

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ՏՆԱՅԻՆ․

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253-բ,գ.

բ) AD = 2,25 դմ, CD = 1,75 դմ, AB = ?

AC = AD + CD = 4 դմ
AB² = AC * AD
AB² = 4 * 2,25 = 9
AB = √9 = 3 սմ

Պատ․ AB = 3 սմ

գ) AD = 1 մ, CD = 2 մ

AC = AD + CD = 3 մ
AB² = AC * AD
AB² = 3
AB = √3 սմ ≈ 1,73 սմ

Պատ․ AB ≈ 1,73 սմ

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255.

AD = x
AC = 2 ¹/₄x = 2,25x

AB² = AC * AD
AB² = 2,25x * x = 2,25x²
AB = √2,25x² = 1,5x
AC/AB = 2,25x/1,5x = 1,5

Պատ․ AC = 1,5AB

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257.

AB = 20 սմ, AC = 40 սմ, OM = 8 սմ, r = ?

AB² = AC * AD
20² = 40 * AD
AD = 400/40 = 10 սմ
CD = 40 – 10 = 30 սմ
DM = 30/2 = 15 սմ

OD² = DM² + OM²
OD² = 15² + 8² = 225 + 64 = 289
OD = √289 = 17 սմ
r = OD = 17 սմ

Պատ․ r = 17 սմ 

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261.

AB = 4 սմ, AC = 8 սմ, OM = 12 սմ, AO = ?

AB² = AC * AD
4² = 8 * AD
AD = 4²/8 = 16/8 = 2 սմ

CD = AC – AD = 8 – 2 = 6 սմ
DM = CD/2 = 6/2 = 3 սմ
AM = AD + DM = 2 + 3 = 5 սմ

AO² = AM² + OM² = 5² + 12² = 25 + 144 = 169
AO = √169 = 13 սմ

Պատ․ AO = 13 սմ

13.2.1

2,5/0,75 = h/6
10/3 = h/6
h = 20 մ

Պատ․ h = 20 մ

14.1.8

α = 25°
α + γ = ?

α = 25°
γ = α = 25°
α + γ = 25° + 25° = 50°

Պատ․ α + γ = 50°

14.1.12

α + γ = 150°

α = γ = 150°/2 = 75°
90° – 75° = 15°

Պատ․ 15°

15.1.8

α = 60°
β = 30°
V₁ = 3 * 10⁸ մ/վ
V₂ = ?

sin α/sin β = V₁/V₂
sin 60°/sin 30° = (3 * 10⁸ մ/վ)/V₂
(3 * 10⁸ մ/վ)/V2 = 0,5√3/0,5 = √3
V₂ = (3 * 10⁸ մ/վ)/√3 = √3 * 10⁸ մ/վ ≈ 1,7 * 10⁸ մ/վ

Պատ․ V₂ ≈ 1,7 * 10⁸ մ/վ

15.1.12

n = 1,41

β = 30°
α = ?

sin α/sin β = n
sin α/sin 30° = 1,41
sin α = 1,41 * sin 30° = 1,41 * 0,5 = 0,705
α = sin^-1 0,705 ≈ 45°

Պատ․ α ≈ 45°

15.1.14

α = 45°
β = 30°
V₁ = 3 * 10⁸ մ/վ
V₂ = ?

sin α/sin β = V₁/V₂
sin 45°/sin 30° = (3 * 10⁸ մ/վ)/V₂
(3 * 10⁸ մ/վ)/V₂ = (1/√2)/0,5 = 2/√2
V₂ = (3 * 10⁸ մ/վ)/(2/√2) = 1,5√2 * 10⁸ մ/վ ≈ 2,1 * 10⁸ մ/վ

Պատ․ 2,1 * 10⁸ մ/վ

15.1.18

α = 60°
β = ?

n_ջուր = 1,33
sin α/sin β = n
sin 60°/sin β = 1,33
sin β = sin 60°/1,33 = (√2/2)/1,33
sin β = √2/2,66 ≈ 0,53

β = sin^-1 0,53 ≈ 32°

Պատ․ 32°