SOLUTION: The area of a triangle is equal to 41 sq cm and two of its sides measure 17 cm and 16 cm. Find the possible measure of the included angle in degrees of the given sides.

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Question 1150862: The area of a triangle is equal to 41 sq cm and two of its sides measure 17 cm and 16 cm. Find the possible measure of the included angle in degrees of the given sides.
Found 5 solutions by rothauserc, ikleyn, MathTherapy, Alan3354, jim_thompson5910:
Answer by rothauserc(4718)   (Show Source):
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Area of a triangle = (1/2) * base * height,
:
the given area of the triangle is 41 square cm, therefore
:
41 = (1/2) * base * height
:
82 = base * height
:
the factors of 82 are 1, 2, 41, 82
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the sum of of any two sides of a triangle must be greater than the third, therefore
:
possible value for the third side is 2 since 17 +16 < 41 and 17 +16 < 82 and 1+16 = 17
:
using the law of cosines, we have
:
cos(c) = (A^2 +B^2 -C^2)/2AB where c is the angle between sides A, B
:
cos(c) = (17^2 +16^2 -2^2 -2 * 17 * 16)/(2 * 17 * 16) = −0.0055
:
cos^-1 −0.0055 = 90.3151
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therefore, possible value for c is 90.3151 degrees
:

Answer by ikleyn(52802)   (Show Source):
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.

Use the formula for the area of a triangle S = where "a" and "b" are two any side lengths of the triangle and is the concluded angle between them. From the formula, = = = 0.301471. Hence, = = 0.309634 radians = 17.74 degrees. The other possible value for the angle is 180 degrees MINUS 17.74 degrees = 162.24 degrees. ANSWER. At given conditions, there are two possible values for the angle between the sides 17.74 degrees OR 162.24 degrees (approximately).

Solved.

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Ignore the solution and the answer by other person, since it is WRONG and IRRELEVANT.

The fact that it is wrong, you can easily check by calculating the area of the triangle with the angle of 90.3 degrees between the side.

You can replace then sin(90.3 degree) by 1; it gives you the value for the area of the triangle = 8*17 = 136 square centimeters,

which is VERY FAR from the given value of 41 square centimeters.

Answer by MathTherapy(10552)   (Show Source):
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The area of a triangle is equal to 41 sq cm and two of its sides measure 17 cm and 16 cm. Find the possible measure of the included angle in degrees of the given sides.

The other person is WRONG!! The INCLUDED angle is NOT 90.3151^o.
Formula for the area of a NON-RIGHT triangle: * sin ∡C, with ∡C being the INCLUDED angle, or the angle between the 2 GIVEN sides
* sin ∡C ------ Substituting 41 for A, 17 and 16, for a and b, respectively
* sin ∡C
sin ∡C
∡C, or INCLUDED angle =
This is the INCLUDED-ANGLE measurement if this angle is the SMALLEST of the 3, which would make the unknown 3rd side, the SHORTEST of the 3,
and which would be a length that's > 1 cm, but < 16 cm.

Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website!
There's a place called Squabbletown in California.

Answer by jim_thompson5910(35256)   (Show Source):
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There are already some great answers here. So I don't have much to add other than provide a visual of what the triangles look like.

Image generated by GeoGebra (free graphing software).
The angle answers are shown in red, which are approximate values.

Triangles ABC and DEF have the same area of 41 square cm.