SOLUTION: A rhombus has side length 32 centimeters and one of its angles has measure 50 degrees. Find the exact length of the shorter diagonal. (Hint: The exact length will involve a trigono

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Question 969331: A rhombus has side length 32 centimeters and one of its angles has measure 50 degrees. Find the exact length of the shorter diagonal. (Hint: The exact length will involve a trigonometric ratio.) I'm not sure what to do. But the answer is 64 sin 25 degrees centimeters. Thank you for your help!! I'm thinking that since 25 is half of 50, it could have something to do with angle bisectors and 32 is half of 64, but I'm not sure how that has something to do with it. Thank you for your help!
Answer by solver91311(24713) About Me  (Show Source):
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The diagonals of a rhombus are always perpendicular bisectors of each other and always bisect the vertex angles of the rhombus. Since adjacent angles of any parallelogram are supplementary, and we are given one of the angles as 50 degrees, the other angle has to be 180 minus 50 or 130 degrees.

Since the diagonals are perpendicular, by constructing them you have created four right triangles. In this case, each of your right triangles has a hypotenuse of 32 and acute angles of 25 degrees and 65 degrees. The measure of the hypotenuse multiplied times the sine of 25 degrees which is the angle of the triangle opposite the short leg, gives you the measure of the short leg.

Since the diagonals are bisectors, the short leg of one of the right triangles is exactly half of the measure of the short diagonal.



John

My calculator said it, I believe it, that settles it