SOLUTION: Hi, I've spent so much time on this one problem and still can't figure it out! If you could help me out, it would be really appreciated! A rectangle has a diagonal 20 inches lon

Algebra ->  Points-lines-and-rays -> SOLUTION: Hi, I've spent so much time on this one problem and still can't figure it out! If you could help me out, it would be really appreciated! A rectangle has a diagonal 20 inches lon      Log On


   

Question 857594: Hi, I've spent so much time on this one problem and still can't figure it out! If you could help me out, it would be really appreciated!
A rectangle has a diagonal 20 inches long that forms angles of
60° and 30° with the sides. Find the perimeter of the rectangle.
Thanks
Found 2 solutions by josgarithmetic, josmiceli:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
That makes an important-to-know special triangle. The remaining angle is 90 degrees, which you would expect, knowing that the diagonal is in a rectangle. Each triangle formed is half of an equilateral triangle. The diagonal of your rectangle is one of the equilateral triangles equal sides. The short leg of the special triangle is HALF the diagonal, or 10 inches.

The long leg is the same as the long side of the rectangle. Find it using Pythagorean Theorem for Right Triangle:
y%5E2%2B10%5E2=20%5E2
y%5E2=400-100=300
y=sqrt%28300%29
highlight_green%28y=10%2Asqrt%283%29%29-----the long side of the rectangle.
Remember, short leg, the shorter side of the rectangle, is 10.

I leave the perimeter part to you.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
The diagonal splits the rectangle into 2
30-60-90 triangles. What you need to know
for this problem is the proportions of the sides.
(1) side opposite 30 degree angle: +1+
(2) side opposite 60 degree angle: +sqrt%283%29+
(3) side opposite 90 degree angle: +2+
( note that +1%5E2+%2B+sqrt%283%29%5E2+=+2%5E2+ )
---------------------------------------------
Now you can say:
+a+%2F+20+=+1+%2F+2+
( short side of given triangle ) / ( diagonal of given triangle ) =
( ratio of short side / diagonal of all 30-60-90 triangles )
Multiply both sides by +20+
+a+=+10+
--------------------
And also:
+b+%2F+20+=+sqrt%283%29+%2F+2+
( long side of given triangle ) / ( diagonal of given triangle ) =
( ratio of long side / diagonal of all 30-60-90 triangles )
Again, multiply both sides by +20+
+b+=+10%2Asqrt%283%29+
----------------------------
The perimeter, P, is:
+P+=+2a+%2B+2b+
+P+=+2%2A10+%2B+2%2A10%2Asqrt%283%29+
+P+=+20+%2B+20%2Asqrt%283%29+
Hope this isn't confusing