How do you find the measure of each of the angles of a triangle given the measurements of the sides are 12, 20, 22?

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How do you find the measure of each of the angles of a triangle given the measurements of the sides are 12, 20, 22?

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Answer 1 · 2016-12-06T13:59:41

Use the Law of Cosines as demonstrated below.

Explanation:

Let $∠ A$ $/_A$ be the angle opposite side $| a | = 22$ $abs(a)=22$
and $∠ B$ $/_B$ be the angle opposite side $| b | = 20$ $abs(b)=20$
and $∠ C$ $/_C$ be the angle opposite side $| c | = 12$ $abs(c)=12$

The Law of Cosines tells us that
$XXX {| a |}^{2} = {| b |}^{2} + {| c |}^{2} - 2 | b | | c | cos (A)$ $color(white)("XXX")abs(a)^2=abs(b)^2+abs(c)^2-2abs(b)abs(c)cos(A)$
or (in a form more useful for this problem):
$XXX cos (∠ A) = \frac{{| b |}^{2} + {| c |}^{2} - {| a |}^{2}}{2 | b | | c |}$ $color(white)("XXX")cos(/_A)= (abs(b)^2+abs(c)^2-abs(a)^2)/(2abs(b)abs(c))$
or...
$XXX ∠ A = arccos (\frac{{| b |}^{2} + {| c |}^{2} - {| a |}^{2}}{2 | b | | c |})$ $color(white)("XXX")/_A = "arccos"((abs(b)^2+abs(c)^2-abs(a)^2)/(2abs(b)abs(c)))$

Using the given values (and a calculator)
$XXX ∠ A \approx 1.445468$ $color(white)("XXX")/_A ~~ 1.445468$ (radians)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Similarly, we can find:
$XXX ∠ B \approx 1.124289$ $color(white)("XXX")/_B~~1.124289$ (radians)
and
$XXX ∠ C \approx 0.571835$ $color(white)("XXX")/_C~~0.571835$ (radians)

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How do you find the measure of each of the angles of a triangle given the measurements of the sides are 12, 20, 22?

1 Answer

Explanation:

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