Mibvunzo yeMienzaniso neKukurukurirana kweMadenderedzwa neMatagoni
Madenderedzwa ne tangents inyaya mbiri dzinokurukurwa kakawanda mumasvomhu, kunyanya padanho rechikoro chesekondari. Kunzwisisa pfungwa uye kushandiswa kwe tangents kumadenderedzwa kwakakosha pakuwedzera ruzivo rwako rwe geometry. Chinyorwa chino chichapa mienzaniso yezvinetso nehurukuro pamusoro pemadenderedzwa ne tangents kuti vaverengi vanzwisise zvakadzama.
Nhanganyaya kuDzidziso yeDzungu neTangents
Denderedzwa
Denderedzwa iboka remapoinzi ari mudenderedzwa ari kure zvakaenzana kubva panzvimbo yakatarwa inonzi pakati pedenderedzwa. Daro iri rakagadziriswa rinozivikanwa se radius yedenderedzwa. Pamasvomhu, denderedzwa rinogona kutsanangurwa ne equation:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
apo \((a, b)\) ari makoronesheni ari pakati pedenderedzwa uye \(r\) ari radius.
Tangent
Tangenti kudenderedzwa mutsetse unobata denderedzwa panzvimbo imwe chete chaiyo. Nzvimbo iyi inonzi poindi yetangenti. Hunhu hukuru hwetangenti ndehwekuti yakatarisana neradius inotorwa kubva pakati pedenderedzwa kusvika panzvimbo yetangenti.
Mibvunzo yemuenzaniso nekukurukurirana
Mubvunzo 1: Kuona Equation yeTangent Line
Mubvunzo:
Kupiwa denderedzwa rine pakati pa \( (2, 3) \) uye radius 5. Sarudza equation yemutsetse wetangent kune denderedzwa panzvimbo \( P \) ine ma coordinates \( (5, 7) \).
Kukurukurirana:
Danho 1: Iva nechokwadi chekuti poindi \( P \) iri mudenderedzwa.
Kuti uone kana \( P (5, 7) \) iri padenderedzwa rine pakati \( (2, 3) \) uye radius \( 5 \), chinja macoordinates e \( P \) mu equation yedenderedzwa:
\[ (x – 2)^2 + (y – 3)^2 = 5^2 \]
\[ (5 – 2)^2 + (7 – 3)^2 = 25 \]
\[ 3^2 + 4^2 = 25 \]
\[ 9 + 16 = 25 \]
Sezvo kuenzana kuri kwechokwadi, pfungwa \( P \) iri padenderedzwa.
Danho rechipiri: Sarudza kuti radius iri kupfuura nepakati pe \( (2, 3) \) uye \( (5, 7) \):
\[ m_{radius} = \frac{y_2 – y_1}{x_2 – x_1} = \frac{7 – 3}{5 – 2} = \frac{4}{3} \]
Danho rechitatu: Kuyerera kwemutsetse we tangent kwakatarisana ne gradient ye radius (gradient yechigadzirwa i -1):
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]
Danho rechina: Sarudza equation yemutsetse wetangent uchishandisa poindi \( P (5, 7) \):
\[ y – y_1 = m (x – x_1) \]
\[ y – 7 = -\frac{3}{4} (x – 5) \]
Nyoresa:
\[ y – 7 = -\frac{3}{4}x + \frac{15}{4} \]
\[ 4y – 28 = -3x + 15 \]
\[ 3x + 4y – 43 = 0 \]
Saka, equation yemutsetse wetangent ndeiyi:
\[ 3x + 4y – 43 = 0 \]
Mubvunzo 2: Kuziva Poindi yeTangency kubva kuLine Equation
Mubvunzo:
Kupiwa denderedzwa rine equation \( x^2 + y^2 = 25 \) uye mutsetse \( y = \frac{3}{4}x + 2 \). Sarudza poindi yekubatana pakati pemutsetse nedenderedzwa.
Kukurukurirana:
Danho 1: Isa equation yemutsetse mu equation yedenderedzwa:
Equation yedenderedzwa:
\[ x^2 + y^2 = 25 \]
Isa \( y = \frac{3}{4}x + 2 \) muequation yedenderedzwa:
\[ x^2 + \left(\frac{3}{4}x + 2\right)^2 = 25 \]
\[ x^2 + \left(\frac{9}{16}x^2 + \frac{12}{4}x + 4 \right) = 25 \]
\[ x^2 + \frac{9}{16}x^2 + \frac{6}{2}x + 4 = 25 \]
\[ x^2 + \frac{9}{16}x^2 + 3x + 4 = 25 \]
Danho rechipiri: Nyoresa equation:
\[ 16x^2 + 9x^2 + 48x + 64 = 400 \]
\[ 25x^2 + 48x + 64 – 400 = 0 \]
\[ 25x^2 + 48x – 336 = 0 \]
Danho rechitatu: Kutsvaga midzi uchishandisa fomura yequadratic:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
\[ a = 25, b = 48, c = -336 \]
\[ x = \frac{-48 \pm \sqrt{48^2 – 4 \cdot 25 \cdot (-336)}}{2 \cdot 25} \]
\[ x = \frac{-48 \pm \sqrt{2304 + 33600}}{50} \]
\[ x = \frac{-48 \pm \sqrt{35904}}{50} \]
\[ x = \frac{-48 \pm 189.501}{50} \]
Kusarudza \( x \) inoshanda zvichibva pane tangency point (imwe chete \( x \) ndiyo ichaburitsa tangency point):
\[ x = \frac{141.501}{50} \inenge 2.83 \]
\[x \inenge 2.83 \]
Danho rechina: Isa \( x \) mu equation yemutsetse kuti uwane \( y \):
\[ y = \frac{3}{4}(2.83) + 2 \]
\[ y \inenge 2.12 + 2 \]
\[ y \inenge 4.12 \]
Saka, poindi yekubatana pakati pemutsetse \( y = \frac{3}{4}x + 2 \) nedenderedzwa \( x^2 + y^2 = 25 \) ndiyo \( (2.83, 4.12) \).
Mhedziso
Kuziva pfungwa dzemadenderedzwa ne tangents kunosanganisira kunzwisisa nheyo dze geometry uye kugona kugadzirisa matambudziko uchishandisa masvomhu ekuenzanisa. Matambudziko akaita seari pamusoro apa anobatsira vadzidzi kudzidzira kushandisa dzidziso mumamiriro ezvinhu akajeka. Nekudzidzira kwakasimba, vadzidzi vanotarisirwa kunzwisisa nekugadzirisa matambudziko zviri nyore.
