Dan is making a border for a triangular placemat...

Question

He wants it to have side lengths of a= 8 inch and b= 4 in. If m(angle)A=50, how many different placemats can Dan make?

a) 0

b) 1

Answer 1

To determine whether the answer is 0 or 1, try to figure out what the remaining angles and sides must be to complete the triangle.  If you able to arrive at a legitimate triangle, then the answer would of course be one.  If not, then the answer would be 0.

 

To start off, You can use the law of sines to determine the measure for angle B.

 

sin(A)/a = sin(B)/b

 

so,

 

sin(50 deg)/8 = sin(B)/4

 

Cross multiply to get

 

4sin(50deg) = 8sin(B)

 

sin(B) = 0.5sin(50 deg)

 

Take the inverse sine of both sides to determine the measure of angle B (you should get ~22.52 deg)

 

From this you can determine angle C (180 - 50 - 22.52) to be ~105.48

 

Then use the law of sines again to find c.

 

sin(A)/a = sin(C)/c

 

sin(50deg)/8 = sin(105.48)/c

 

Solve the same way we did above, you should get c ~10.06

 

So to review a = 8, A = 50 deg, b = 4, B = 22.52 deg, c = 10.06, C = 105.48.

 

If you draw this out, the smallest side is across from the smallest angle, and the biggest side is across from the biggest angle, as it should be.  Therefore, we have found a triangle that meet the criteria, making the answer to the question b.