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7.3: Mistari ya Perp

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    164591

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    Ufafanuzi: Perpendicular

    Mstari miwili tofauti\(l\) na\(q\) ni perpendicular\(l ⊥ q\), imeandikwa, ikiwa makutano yao huunda pembe nne za kulia au pembe kwa kipimo\(90^{\circ}\). Miteremko ya mistari ya perpendicular\(l\) na\(q\) ni hasi za usawa. Hiyo ni,

    \[m_l = −\dfrac{1}{m_q} \nonumber \]

    na

    \[m_q = − \dfrac{1}{m_l} \nonumber \]

    Mfano Template:index

    Kuamua kama mistari iliyotolewa ni perpendicular. Mstari\(l\) unaopita kupitia pointi\((0, 1)\) na\((1, 3)\), na mstari\(q\) unaopita kupitia pointi\((−1, 4)\) na\((5, 1)\).

    Suluhisho

    Kuamua kama mistari ni perpendicular, kwanza kupata mteremko wao kwa kutumia mteremko wa formula line. Mteremko wa mstari\(l\)\(m_l\),, kwamba hupita kupitia pointi\((0, 1)\) na\((1, 3)\) ni,

    \(\begin{array}s m_l &= \dfrac{3 − 1}{1 − 0} \\ &= \dfrac{2}{1} \\ &= 2 \end{array}\)

    mteremko wa mstari\(q\),\(m_q\), kwamba hupita kwa njia ya pointi\((−1, 4)\) na\((5, 1)\), ni

    \(\begin{array}s m_q &= \dfrac{1 − 4}{5 − (-1)} \\ &= \dfrac{-3}{6} \\ &= \dfrac{-1}{2} \end{array}\)

    Sasa, mistari\(l\) na\(q\) ni perpendicular kama na tu kama:

    \(m_l = −\dfrac{1}{m_q} \text{ and } m_q = −\dfrac{1}{m_l}\)

    \(m_l = 2\)na\(m_q = −\dfrac{1}{m_l} = −\dfrac{1}{2}\). Kwa hiyo, mteremko wa mistari ni usawa mbaya, hivyo inaweza kuhitimishwa kuwa mistari\(l\) na\(q\) ni mistari ya perpendicular.

    Mfano Template:index

    Pata mteremko wa mstari perpendicular kwa mstari\(l\) unaopita kupitia pointi\((−3, 0)\) na\((3, 4)\).

    Suluhisho

    Anza kwa kutafuta mteremko wa mstari\(l\) unaopita kupitia pointi\((−3, 0)\) na\((3, 4)\), kwa kutumia mteremko wa fomu ya mstari. Hivyo,

    \(\begin{array} s m_l &= \dfrac{y_2 − y_1}{x_2 − x_1} \\ &= \dfrac{4 − 0}{3 − (−3)} \\ &= \dfrac{4}{6} \\ &= \dfrac{2}{3} \end{array}\)

    Mstari wowote unaozingatia mstari\(l\) lazima uwe na mteremko ambao ni hasi usawa wa mteremko wake. \(m_l = \dfrac{2}{3}\)Tangu wakati huo mteremko wa mstari perpendicular line\(l\) lazima\(m = −\dfrac{3}{2}\)

    Zoezi Template:index

    Kuamua kama mistari iliyotolewa ni perpendicular.

    1. Mstari\(l\) unaopita kupitia pointi\((0, 4)\)\((5, 3)\) na na mstari\(q\) unaopita kupitia pointi\((1, 5)\) na\((−1, −5)\).
    2. Mstari\(l\) unaopita kupitia pointi\((−2, −5)\)\((1, 7)\) na na mstari\(q\) unaopita kupitia pointi\((−4, 1)\) na\((−3, −3)\).
    Zoezi Template:index

    Pata mteremko wa mstari perpendicular kwa:

    1. Line\(l\) kwamba hupita kwa njia ya pointi\((4, 2)\) na\((−1, −2)\).
    2. Line\(q\) kwamba hupita kwa njia ya pointi\((7, −8)\) na\((9, 1)\).