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Query

2.4: Mifano iliyowekwa

  • Page ID
    164622
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    Katika sehemu hii, tumia fomu ya umbali\(d = \sqrt{(x_2 − x_1) ^2 + (y_2 − y_1) ^2}\) ili kupata urefu wa makundi ya mstari.

    Kumbuka: pointi tatu\(A\)\(B\), na\(C\) ni collinear, au kwa maneno mengine, pointi tatu ziko kwenye mstari huo, ikiwa jumla ya urefu wa makundi yoyote mawili ya mstari kuunganisha pointi, ni sawa na urefu wa sehemu iliyobaki ya mstari. Hiyo ni,\(AB + BC = AC\) au,\(AB + BC = AC\) au,\(AB + AC = BC\) au\(AC + BC = AB\).

    Kuamua kama pointi tatu zilizopewa ni collinear.

    \(A(10, −4)\quad B(8, −2) \quad C(2, 4)\)

    Suluhisho

    Kwanza kupata makundi\(AB\),\(BC\), na\(AC\). Ili kufanya hivyo, tafuta umbali kati ya pointi\(A\) na\(B\),\(B\) na\(C\),\(A\) na\(C\).

    \(\begin{aligned} \text{Segment AB }&=\text{ The distance between point A and Point B } \\ &= \sqrt{(8 − 10)^2 + [−2 − (−4)]^2} \\ &= \sqrt{(−2)^2 + (2)^2} \\&= \sqrt{ 8}\\&= 2\sqrt{2} \end{aligned}\)

    \(\begin{aligned} \text{Segment BC }&=\text{ The distance between point B and Point C } \\ &= \sqrt{(2 − 8)^2 + [4 − (−2)]^2 }\\ &= \sqrt{(−6)^2 + (6)^2} \\&= \sqrt{ 72 }\\&= 6\sqrt{ 2}\end{aligned}\)

    \(\begin{aligned} \text{Segment AC }&=\text{ The distance between point A and Point C }\\&= \sqrt{(2 − 10)^2 + [4 − (−4)]^2} \\&= \sqrt{(−8)^2 + (8)^2 }\\&= \sqrt{ 128 }\\&= 8\sqrt{ 2}\end{aligned}\)

    Hivyo,

    \(\begin{aligned} AB + BC &= 2\sqrt{ 2} + 6\sqrt{ 2 }\\&= 8\sqrt{ 2 } \\&= AC \end{aligned}\)

    Tangu Hivyo,\(AB + BC = AC\) basi pointi tatu ni collinear.

    1. Kuamua kama pointi zifuatazo ni collinear.
      1. \(A(4,-1)\quad B(5,-2) \quad C(1,2)\)
      2. \(A(2,-2)\quad B(3,1)\quad C(2,1)\)