Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 6 Herrega kutaa 11: Tiraanisfoormeeshiniiwwan Pileenii

Wanti fuula kana irra jiru hundi boqonnaa 6 kitaaba Herrega kutaa 11 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

4
Kaardii: 10
Gaaffii shaakalaa boqonnaa 6: 313–354
Fuula kitaabaa boqonnaa 6

Boqonnaan 6 maal haguuga

1.Translation
2.Reflection
3.Rotation
4.Applications

Yaadannoo gabaabaa boqonnaa 6

Kaardiiwwan fuula Herrega kutaa 11 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Sarara y = mx Irratti CalaqqisiisuuFoormulaa

\begin{cases} x' = x\cos(2\theta) + y\sin(2\theta) \\ y' = x\sin(2\theta) - y\cos(2\theta) \end{cases}

\begin{cases} x' = x\cos(2\theta) + y\sin(2\theta) \\ y' = x\sin(2\theta) - y\cos(2\theta) \end{cases}

$x'$: x coordinate qabxii calaqqisee $dimensionless$
$y'$: y coordinate qabxii calaqqisee $dimensionless$
$x$: x coordinate qabxii jalqabaa $dimensionless$
$y$: y coordinate qabxii jalqabaa $dimensionless$
$\theta$: kofa sarara y = mx bakka m = tan theta ta'etti $\text{rad}$

Yoom akka itti fayyadamtu

Qabxii (x, y) kamiyyuu sarara wiirtuu jalqabaa keessa darbu kan rirriitun isaa m = tan(theta) ta'e irratti calaqqisiisuuf fayyadami.

Of eeggadhu

Foormulaan kun saayinii fi koosaayinii keessatti 2 theta fayyadama malee theta qofa miti.

Grade 11 Mathematics, fuula 313-354 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Qabxii Kamiyyuu Irratti NaannessuuFoormulaa

\begin{cases} x' = a + (x - a)\cos\theta - (y - b)\sin\theta \\ y' = b + (x - a)\sin\theta + (y - b)\cos\theta \end{cases}

\begin{cases} x' = a + (x - a)\cos\theta - (y - b)\sin\theta \\ y' = b + (x - a)\sin\theta + (y - b)\cos\theta \end{cases}

$x'$: x coordinate qabxii fakkii $dimensionless$
$y'$: y coordinate qabxii fakkii $dimensionless$
$x$: x coordinate qabxii jalqabaa $dimensionless$
$y$: y coordinate qabxii jalqabaa $dimensionless$
$a$: x coordinate wiirtuu naanna'aa $dimensionless$
$b$: y coordinate wiirtuu naanna'aa $dimensionless$
$\theta$: kofa naanna'aa $\text{rad}$

Yoom akka itti fayyadamtu

Qabxiin tokko wiirtuu kamiyyuu (a, b) irratti kofa kennameen kallattii faallaa sa'aatiin yommuu naanna'u fakkii isaa barbaaduuf fayyadami.

Of eeggadhu

Hojii tiiriigonoomeetirii dura qindaa'inoota wiirtuu (a, b) hir'isuu fi dhuma irratti deebistanii ida'uu hin dagatinaa.

Grade 11 Mathematics, fuula 313-354 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Wiirtuu Jalqabaa Irratti NaannessuuGabatee wabii90^° · 180^° · 270^° · +1

Kofa Naanna'aa	Fakkii (x, y)
$90^\circ$	$(-y, x)$
$180^\circ$	$(-x, -y)$
$270^\circ$	$(y, -x)$
$360^\circ$	$(x, y)$

Yoom akka itti fayyadamtu

Wiirtuu jalqabaa (0, 0) irratti kallattii faallaa sa'aatiin kofoota beekamoodhaan saffisaan shallaguuf gabatee kana fayyadami.

Of eeggadhu

Naanna'aan kallattii sa'aatii faallaa sa'aatii digrii 360 irraa kofa kenname hir'isuun walqixa.

Grade 11 Mathematics, fuula 313-354 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Calaqqeewwan Beekamoo Dirrii IrrattiGabatee wabiix axis (y = 0) · y axis (x = 0) · y = x · +1

Sarara Calaqqee	Fakkii (x, y)
$x\text{ axis } (y = 0)$	$(x, -y)$
$y\text{ axis } (x = 0)$	$(-x, y)$
$y = x$	$(y, x)$
$y = -x$	$(-y, -x)$

Yoom akka itti fayyadamtu

Qabxiin tokko sararoota beekamoo irratti yommuu calaqqisu qindaa'inoota fakkii isaa argachuuf gabatee kana fayyadami.

Of eeggadhu

Sarara y = -x irratti calaqqisiisuun bakka wal jijjiiree lamaan isaaniyyuu negaatiivii taasisa፡ sarara y = x irratti garuu bakka qofa wal jijjiiru.

Grade 11 Mathematics, fuula 313-354 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 6

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

According to the vector summary, what is the definition of a zero vector (null vector)?

A.A vector whose magnitude is zero and whose direction is indeterminate.
B.A vector whose magnitude is one and whose direction is fixed.
C.A vector whose magnitude is undefined and whose direction is zero.
D.A vector whose magnitude is negative and direction is horizontal.

Deebii fi sababa agarsiisi

Deebii: A A vector whose magnitude is zero and whose direction is indeterminate.

Sababa

By definition, a zero vector or null vector is a vector whose magnitude is zero and whose direction is indeterminate.

Gaaffii 2

If a circle is centred at the origin with radius r, which equation represents the tangent line to the circle at the point (x₁, y₁)?

A.x x1 + y y1 = r^2
B.x x1 - y y1 = r^2
C.(x + x1) + (y + y1) = r
D.x^2 x1 + y^2 y1 = r^2

Deebii fi sababa agarsiisi

Deebii: A x x1 + y y1 = r^2

Sababa

When a circle is centred at the origin, the general tangent line formula simplifies directly to x x₁ + y y₁ = r².

Gaaffii 3

Transformations that preserve shape, size, and the distance between any two points are classified as which type of motion?

A.Rigid motion
B.Non-rigid motion
C.Scalar projection
D.Variable expansion

Deebii fi sababa agarsiisi

Deebii: A Rigid motion

Sababa

A rigid motion is a transformation of the plane that preserves the shape, size, and distance between any two points.

Gaaffii 4

According to the properties of cross products, what is the cross product of any vector u with itself (u × u)?

A.0
B.|u|^2
C.1
D.-u

Deebii fi sababa agarsiisi

Deebii: A 0

Sababa

The cross product of any vector with itself is always the zero vector, so u × u = 0.

Gaaffii 5

A student in Dire Dawa throws a ball with an initial velocity of 70 m/s at an angle of 35° horizontally. What are the horizontal and vertical components of the velocity?

A.Horizontal: 57.34 m/s, Vertical: 40.15 m/s
B.Horizontal: 40.15 m/s, Vertical: 57.34 m/s
C.Horizontal: 70.00 m/s, Vertical: 35.00 m/s
D.Horizontal: 49.50 m/s, Vertical: 49.50 m/s

Deebii fi sababa agarsiisi

Deebii: A Horizontal: 57.34 m/s, Vertical: 40.15 m/s

Sababa

The velocity vector is v = 70 cos(35°) i + 70 sin(35°) j. Simplifying gives v = 57.34 i + 40.15 j, so the horizontal component is 57.34 m/s and the vertical component is 40.15 m/s.

Gaaffii 6

What is the equation of the tangent line through the point R(3, 4) on the circle x² + y² = 25?

A.3x + 4y = 25
B.4x + 3y = 25
C.3x - 4y = 25
D.4x - 3y = 25

Deebii fi sababa agarsiisi

Deebii: A 3x + 4y = 25

Sababa

Using the formula x x₁ + y y₁ = r² for a circle centred at the origin with radius 5, substituting (3, 4) gives 3x + 4y = 25.

Gaaffii 7

Consider the vectors u = i + 5k and v = 10i + 3j - 2k. Are these vectors orthogonal?

A.Yes, because their dot product is 1(10) + 0(3) + 5(-2) = 0.
B.No, because their dot product is 10 + 3 - 10 = 3.
C.No, because their cross product is 0.
D.Yes, because they have the same magnitude.

Deebii fi sababa agarsiisi

Deebii: A Yes, because their dot product is 1(10) + 0(3) + 5(-2) = 0.

Sababa

Two vectors are orthogonal if their scalar (dot) product is zero. Here, u · v = (1)(10) + (0)(3) + (5)(-2) = 10 - 10 = 0, so they are orthogonal.

Gaaffii 8

Using vector methods, the diagonals of a parallelogram meet at right angles if and only if the parallelogram is a:

A.rhombus
B.trapezoid
C.scalene quadrilateral
D.general rectangle

Deebii fi sababa agarsiisi

Deebii: A rhombus

Sababa

The dot product of the diagonal vectors (u + v) · (v - u) = |v|² - |u|² equals 0 if and only if |v| = |u|, which means the adjacent sides are equal in length (a rhombus).

Gaaffii 9

What is the equation of the circle whose diameter has endpoints P(-1, -1) and Q(4, 11)?

A.x^2 + y^2 - 3x - 10y - 15 = 0
B.x^2 + y^2 + 3x + 10y - 15 = 0
C.x^2 + y^2 - 3x - 10y + 15 = 0
D.x^2 + y^2 - 4x - 6y - 3 = 0

Deebii fi sababa agarsiisi

Deebii: A x^2 + y^2 - 3x - 10y - 15 = 0

Sababa

The centre is C(3/2, 5) and radius is 13/2. Expanding (x - 3/2)² + (y - 5)² = (13/2)² gives x² + y² - 3x - 10y - 15 = 0.

Gaaffii 10

Find the equation of the tangent line to the circle x² + y² + 10x + 2y + 13 = 0 at the point (-3, 2).

A.2x + 3y = 0
B.3x + 2y = 0
C.2x + 3y = 13
D.5x + y = -13

Deebii fi sababa agarsiisi

Deebii: A 2x + 3y = 0

Sababa

Completing the square gives centre (-5, -1) and r² = 13. Using the tangent formula (x - x₀)(x₁ - x₀) + (y - y₀)(y₁ - y₀) = r² gives (x + 5)(2) + (y + 1)(3) = 13, which simplifies to 2x + 3y = 0.

Appiin boqonnaa kana irratti gaaffii 18 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.

Fuula kitaabaa boqonnaa 6

Boqonnaan 6 Maxxansa Ministeera Barnootaa keessatti fuula 313 hanga fuula 354 ti.

Maxxansa Ministeera Barnootaa, Ingiliffaa, Fuula 494

PDF buufadhaa

Yaadannoo gabaabaa Herrega kutaa 11 hunda

Boqonnaa kana appii keessatti shaakali

Temari boqonnaa kana gara shaakala yeroon daangeffametti jijjiira; utuu hojjettuu ni sirreessa; wanti si hin galinis barsiisaa AI tiin ni ibsama.

Bilisaan jalqabi

Temari.et appii Android qaba; Google Play irratti bilisaan argama.

Waa'ee appii Temari