math

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Find, correct to the nearest degree, the three angles of the triangle with the given vertices.?

A(2, 0),B(4, 5), C(-3,2)

ANGLE- CAB
ANGLE- ABC
ANGLE- BCA
In degrees.

I got the vectors AB <,5>
BC <1,3>
AC <-1,2>

and I know the equation b.c= |b| |c| cos t

  • math -

    I have no idea how you were taught to calculate vectors.
    For any two points P(a,b) and Q(c,d)
    vector PQ = < c-a, d-b>
    I got
    vector AB = <2,5>
    vector BC = <-7,-3) and
    vector AC = <-5,2>

    let angle CAB = Ø
    then vector AC . vector AB = |AC| |AB| cosØ
    <-5,2> . <2,5> = | <-5,2>| |<2,5>| cosØ
    -10+10 = √29 √29 cosØ
    cosØ = 0
    Ø = 90°

    well, that was obvious from looking at the two vectors, their dot product is zero, so they are perpendicular

    for angle ABC
    <-3,-7> . <-5,-2> = √58 √29 cos B
    15+14 = √58√29cosB
    cos B =.707106...
    B = 45°

    Well, how about that ?

    Suppose we had taken the length of each vector

    |AB = √29
    |AC| = √29
    |BC| = √58

    so it is isosceles, and since
    (√58) ^2 = (√29)^2 + (√29)^2
    so it is also right-angled, as I showed in my first method.

  • math -

    I have no idea how you were taught to calculate vectors.
    For any two points P(a,b) and Q(c,d)
    vector PQ = < c-a, d-b>
    I got
    vector AB = <2,5>
    vector BC = <-7,-3) and
    vector AC = <-5,2>

    let angle CAB = Ø
    then vector AC . vector AB = |AC| |AB| cosØ
    <-5,2> . <2,5> = | <-5,2>| |<2,5>| cosØ
    -10+10 = √29 √29 cosØ
    cosØ = 0
    Ø = 90°

    well, that was obvious from looking at the two vectors, their dot product is zero, so they are perpendicular

    for angle ABC
    <-3,-7> . <-5,-2> = √58 √29 cos B
    15+14 = √58√29cosB
    cos B =.707106...
    B = 45°

    Well, how about that ?

    Suppose we had taken the length of each vector

    |AB = √29
    |AC| = √29
    |BC| = √58

    so it is isosceles, and since
    (√58) ^2 = (√29)^2 + (√29)^2
    so it is also right-angled, as I showed in my first method.

  • math -

    THANK YOU SO MUCH!

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