Zitsanzo za Mafunso Okhudza Magulu ndi Ma Tangents
Mabwalo ndi mutu wofunikira kwambiri mu matrix geometry, komwe malingaliro ozama okhudza mtunda, ma angles, ndi mawonekedwe amawonetsedwa. Lingaliro limodzi lomwe limakambidwa kawirikawiri pamutuwu ndi mzere wozungulira wozungulira. M'nkhaniyi, tikambirana zitsanzo zingapo za mavuto okhudzana ndi mabwalo ndi ma tangents.
Kumvetsetsa Koyambira kwa Ma Circles ndi Tangents
Mzere wozungulira
Bwalo ndi mawonekedwe a geometrical opangidwa ndi gulu la mfundo zonse zomwe zili mu ndege zomwe ndi mtunda wokhazikika kuchokera pamalo ena otchedwa pakati pa bwalo. Mtunda wokhazikika uwu umatchedwa radius ya bwalo.
Tangent
Mzere wozungulira ndi mzere womwe umakhudza bwalo pamalo amodzi. Malo awa amatchedwa malo ozungulira. Ma tangenti ali ndi zinthu zingapo zofunika, kuphatikizapo:
– Mzere wozungulira nthawi zonse umakhala wolunjika ku utali wa bwalo pamalo pomwe mzerewo uli.
– Kutalika kwa tangent kuchokera pa mfundo kunja kwa bwalo kupita ku bwalo ndi kofanana ngati ma tangent awiri atengedwa kuchokera pa mfundoyo.
Mafunso ndi Zokambirana za Zitsanzo
Pansipa tipereka zitsanzo zingapo za mafunso omwe akukambirana mwatsatanetsatane za mabwalo ndi ma tangent.
Chitsanzo Funso 1: Kupeza Kutalika kwa Mzere wa Tangent
Funso:
Popatsidwa bwalo lokhala ndi pakati \(O\) ndi radius \(r = 6 \, \text{cm}\). Kuchokera pamalo \(P\) kunja kwa bwalo lomwe lili 10 cm kuchokera pakati pa bwalo, ma tangent awiri \(PA\) ndi \(PB\) amakokedwa ku bwalo. Werengani kutalika kwa tangent \(PA\).
Kukambirana:
Mu vutoli, tingagwiritse ntchito chiphunzitso cha Pythagorean. Jambulani kansalu \(\triangle OAP\):
– \(OP = 10 \, \text{cm}\) (mtunda kuchokera pa mfundo yakunja kupita pakati pa bwalo)
– \(OA = 6 \, \text{cm}\) (radius ya bwalo)
– \(PA\) ndi mzere wozungulira womwe uyenera kupezeka
\[
OP^2 = OA^2 + PA^2
\]
\[
10^2 = 6^2 + PA^2
\]
\[
100 = 36 + PA^2
\]
\[
PA^2 = 64
\]
\[
PA = \sqrt{64} = 8 \, \text{cm}
\]
Kotero, kutalika kwa mzere wa tangent \(PA\) ndi 8 cm.
Chitsanzo Funso 2: Kupeza Mfundo ya Kukhazikika
Funso:
Kupatsidwa bwalo lokhala ndi equation \((x – 3)^2 + (y – 4)^2 = 25\) ndi mzere \(y = 2x + 1\). Dziwani mfundo ya kugundana pakati pa bwalo ndi mzere.
Kukambirana:
Choyamba, timazindikira pakati ndi utali wa bwalo:
– Pakati \(O(3, 4)\)
– Utali wozungulira \(r = \sqrt{25} = 5\)
Kuti tipeze mfundo ya tangency, tiyeni tiyerekeze kuti mfundo ya tangency ndi \(T(x_1, y_1)\) yomwe ilinso pamzere \(y = 2x + 1\). Kenako:
\[
y_1 = 2x_1 + 1
\]
\(T(x_1, y_1)\) iyeneranso kukwaniritsa equation ya bwalo:
\[
(x_1 – 3)^2 + (y_1 – 4)^2 = 25
\]
Lowetsani \(y_1 = 2x_1 + 1\) mu equation yozungulira:
\[
(x_1 – 3)^2 + ((2x_1 + 1) – 4)^2 = 25
\]
\[
(x_1 – 3)^2 + (2x_1 – 3)^2 = 25
\]
Tiyenera kuwerengera masikweya awiri.
\[
(x_1 – 3)^2 = x_1^2 – 6x_1 + 9
\]
\[
(2x_1 – 3)^2 = 4x_1^2 – 12x_1 + 9
\]
Phatikizani zotsatira zonse ziwiri:
\[
x_1^2 – 6x_1 + 9 + 4x_1^2 – 12x_1 + 9 = 25
\]
\[
5x_1^2 – 18x_1 + 18 = 25
\]
Chotsani 25 kuchokera mbali zonse ziwiri:
\[
5x_1^2 – 18x_1 – 7 = 0
\]
Konzani equation ya quadratic:
\[
x_1 = \frac{18 \pm \sqrt{18^2 + 4 \times 5 \times 7}}{2 \times 5}
\]
\[
x_1 = \frac{18 \pm \sqrt{324 + 140}}{10}
\]
\[
x_1 = \frac{18 \pm \sqrt{464}}{10}
\]
\[
x_1 = \frac{18 \pm 2\sqrt{116}}{10}
\]
\[
x_1 = \frac{18 \pm 2\sqrt{4 \times 29}}{10}
\]
\[
x_1 = \frac{18 \pm 4\sqrt{29}}{10}
\]
\[
x_1 = 1.8 \pm 0.4\sqrt{29}
\]
Werengerani mtengo wa \(y_1\):
Chimene chimakwaniritsa y = 2x + 1:
– Ngati \(x_1 = 1.8 + 0.4\sqrt{29}\), ndiye \(y_1 = 2(1.8 + 0.4\sqrt{29}) + 1\)
– Ngati \(x_1 = 1.8 – 0.4\sqrt{29}\), ndiye \(y_1 = 2(1.8 – 0.4\sqrt{29}) + 1\)
Kuwunika:
Kotero timapeza mfundo ziwiri zomwe zimalumikizana ndi equation ya bwalo ndi mzere umenewo.
Chitsanzo Funso 3: Kudziwa Equation ya Mzere wa Tangent
Funso:
Kupatsidwa bwalo lokhala ndi equation \((x – 2)^2 + (y – 3)^2 = 20\). Dziwani equation ya mzere wozungulira womwe umadutsa mu mfundo \((6, 7)\).
Kukambirana:
Chingwe chozungulira chokhala ndi pakati \((h, k)\) ndi radius \(r\) kuchokera ku mfundo yodziwika yakunja chingapezeke mwa equation:
Mzere wa tangent umadutsa pa mfundo yakunja \((x_1, y_1)\):
\[
(x – 2)(x_1 – 2) + (y – 3)(y_1 – 3) = 20
\]
M'malo mwa mfundo yakunja \((6, 7)\):
\[
(x – 2) (6 – 2) + (y – 3) (7 – 3) = 20
\]
\[
4(x – 2) + 4(y – 3) = 20
\]
\[
4(x – 2 + y – 3) = 20
\]
\[
4x + 2y -20 = 20
\]
\[
4x + 4y -20 = 20
\]
\[
x + y = 5
\]
Equation ya mzere wa tangent ndi:
\[
x + y = 9
\]
Kotero, kusiyana kwa equation ya mzere kudzera pa mfundo ya mzere wozungulira ndi kwakukulu kwambiri ndipo kungasinthe kutengera zotsatira kapena mawonekedwe.
Mapeto
Kukambirana za ma circles ndi ma tangents kumakhudza mbali zingapo zofunika za masamu, kuyambira kugwiritsa ntchito ma formula oyambira monga chiphunzitso cha Pythagorean mpaka kuthetsa ma quadratic equation. Kudzera mu zitsanzo izi, titha kumvetsetsa bwino momwe tingagwiritsire ntchito mfundozi m'mikhalidwe yovuta pang'ono. Tikukhulupirira kuti nkhaniyi yathandiza kupereka chithunzi chomveka bwino cha momwe tingayankhire ndikuthana ndi mavuto okhudzana ndi ma circles ndi ma tangents.
