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2.3: Mfumo wa Umbali

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    164623
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    Sehemu iliyopita ilifundisha jinsi ya kupanga njama katika ndege ya kuratibu mstatili. Sehemu hii inafundisha jinsi ya kupata umbali kati ya pointi zozote mbili kwenye ndege. Kwa mfano, kupata umbali wa pointi\((x_1, y_1)\) na\((x_2, y_2)\) fikiria formula ifuatayo:

    Ufafanuzi: Mfumo wa Umbali

    Umbali d kati ya pointi mbili,\(P_1(x_1, y_1)\) na\(P_2(x_2, y_2)\) katika ndege hutolewa na:

    \(d = \sqrt {(x_2 − x_1) ^2 + (y_2 − y_1)} ^2\)

    Pata umbali kati ya pointi\((−5, 2)\) na\((3, 4)\)

    Suluhisho

    Hebu\(P_1(−5, 2)\) na\(P_2(3, 4)\) uwe na pointi mbili katika ndege na basi\(x_1 = −5\),\(y_1 = 2\),\(x_2 = 3\), na\(y_2 = 4\).

    Kutumia formula ya umbali na maadili yaliyotolewa:

    \(\begin{aligned} d &= \sqrt{(x_2 − x_1) ^2 + (y_2 − y_1) ^2 } \\&= \sqrt{ (3 − (−5))^2 + (4 − 2)^2}\\& = \sqrt{ (3 + 5)^2 + (2)^2 } \\ &= \sqrt{ 8 ^2 + 2^2} \\ &= \sqrt{64 + 4 }\\ &= \sqrt{ 68 } \\&= 2\sqrt{17}\end{aligned}\)

    Kwa hiyo, umbali kati ya pointi mbili zilizopewa ni\(2\sqrt{17}\).

    Kupata umbali kati ya pointi\((−2.5, −1)\) na\((−3, −1.5)\).

    Suluhisho

    Hebu\(P_1(−2.5, −1)\) na\(P_2(−3, −1.5)\) uwe na pointi katika ndege na basi\(x_1 = −2.5\),\(y_1 = −1\),\(x_2 = −3\) na\(y_2 = −1.5\).

    Kisha kutumia formula ya umbali na maadili yaliyotolewa mavuno,

    \(\begin{aligned} d &= \sqrt{(x_2 − x_1) ^2 + (y_2 − y_1) ^2}\\& = \sqrt{[−3 − (−2.5)]^2 + [−1.5 − (−1)]^2 } \\&= \sqrt{ (−3 + 2.5)^2 + (−1.5 + 1)^2} \\&= \sqrt{ (−0.5)^2 + (−0.5)^2 } \\&= \sqrt{ 0.25 + 0.25 }\\ &= \sqrt{0.5 } \\&\approx 0.71 \end{aligned}\)

    Kwa hiyo, umbali kati ya pointi mbili zilizopewa ni takriban 0.71.

    1. Kupata umbali kati\(P_1(−3, −1.5)\) na\(P_2(−2.5, − 1)\). Linganisha jibu kwa jibu katika mfano 2. Ni nini kinachoweza kuhitimishwa?
    2. Pata umbali kati\((−3, 6)\) na\((2, 4)\)
    3. Pata umbali kati ya pointi\(\left( \dfrac{1 }{2} , − \dfrac{10 }{4}\right)\) na\(\left(− \dfrac{14 }{4} , − \dfrac{5 }{2}\right )\)
    4. Kwa nini formula ya umbali hutumiwa?