A triangle has vertices P(2,−1,0),Q(4,1,1),R(4,−5,4). Determine if the triangle is a right triangle (a) Using the Pythagorean theorem; (b) U

Question

A triangle has vertices P(2,−1,0),Q(4,1,1),R(4,−5,4). Determine if the triangle is a right triangle (a) Using the Pythagorean theorem; (b) Using the properties of the dot product.

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Hailey 3 days 2021-09-13T08:32:52+00:00 1 Answer 0

Answers ( )

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    2021-09-13T08:34:34+00:00

    Answer:

    see answers below

    Step-by-step explanation:

    a) using the  Pythagorean theorem

    PQ=   (2,−1,0) – (4,1,1) = (-2,-2,-1)

    QR= (4,1,1) – (4,−5,4) = (0,6,-3)

    RP=    (4,−5,4) – (2,−1,0) = (2,-4,4)

    then

    length 1 = |PQ| = √[(-2)²+(-2)²+(-1)²]= √9 = 3

    length 2 = |QR| = √[(0)²+(6)²+(-3)²]= √45

    length 3 =  |RP| = √[(2)²+(-4)²+(-4)²]= √36 = 6

    then if the triangle is a right triangle

    length 2 = √[(length 1)²+ (length 3)²] = √(3²+6²) = √45

    then our assumption is correct

    b) using the properties of dot product

    PQ * RP  = (-2,-2,-1) * (2,-4,4) = 2*(-2) + (-2)*(-4) + (-1)*4 = -4 + 8 – 4 = 0

    thus PQ is perpendicular to RP . Then the triangle PQR is a right one

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45:7+7-4:2-5:5*4+35:2 =? ( )