SOLUTION: For the following exercises, find the lengths of the missing sides if side a is opposite angle A, side b is opposite angle B,and side c is the hypotenuse. Sin B =1/2 , a=

Question 1200752: For the following exercises, find the lengths of the missing sides if side a
is opposite angle A, side b is opposite angle B,and side c
is the hypotenuse.
Sin B =1/2 , a=20
Thank you
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you have:
sin(B) = 1/2
a = 20
a is opposite angle C
b is oppositge angle B
c is the hypotenuse which is opposite angle C which is the right angle of the right triangle.
if sin(B) = 1/2, then b = 1 and c = 2^2.
by pythagorus, a^2 + b^2 = c^2 gets you a^2 + 1^2 = 2^2
solve for a to get:
a = sqrt(2^2-1^2) = sqrt(4-1) = sqrt(3).
but a = 20, therefore there must a common factor in your right triangle that makes it similar to the larger triangle.
let that common factgor be x.
you get a = 20 in the larger triangle, while a = sqrt(3) in the smaller triangle.
the common factor tells you that 20 = sqrt(3) * x.
solve for x to get x = 20/sqrt(3)
since b in the smaller triangle is 1, then b in the larger triangle is 20/sqrt(3).
since c in the smaller triangle is 2, then c in the larger triangle is 2 * 20/sqrt(3) = 40/sqrt(3).
you get:
a = 20
b = 20/sqrt(3)
c = 40/sqrt(3)
by pythagorus, a^2 + b^2 = c^2 which gets you:
20^2 + (20/sqrt(3))^2 = (40/sqrt(3))^2
evaluate both sides of this equation to get:
533+1/3 = 533+1/3 which is simplified to 1600/3 = 1600/3, confirming the values of a, b, and c are correct.
not sure what you are looking for, but i get:
a = 20
b = 20/sqrt(3)
c = 40/sqrt(3)
since sine(B) = 1/2, then B must be equal to 30 degrees and A must be equal to 60 degrees, while C is 90 degrees.
bsically, you are dealing with similar triangles where the common factor is 20/sqrt(3), which says that the corresponding sides in the larger triangle are equal to the corresponding sides in the smaller triangle multiplied by a factor of 20/sqrt(3)
Answer by ikleyn(52804) (Show Source): You can put this solution on YOUR website!
.

The formulation in this problem is constructed INCORRECTLY.

Traditionally, descriptions in geometric problems go "from the top to the bottom".

The figures are described first, then the descriptions of their elements follow.

So, in the normal description, you should say first that the triangle is a right-angled,
and only after that describe its sides and angles.

All other attempts to builds a description only demonstrate
that the "problem's creator" is unprofessional Math writer.

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