State whether the given measurements determine zero, one, or two triangles.

Question

State whether the given measurements determine zero, one, or two triangles.

B = 84°, b = 28, c = 25

Two
One
Zero

Answer 1

Given b =28,c=25 angle B =84 

 

as micheal said,

if we apply cosine law  we can write cos B =(c^2 +a^2 -b^2)/2ca

 

if you will plug in the values ,we get a quadratic in a

 

.105 =(625+ a^2- 784)/50a

this gives us a^2-.5225 a -159=0

 

if you will check the discriminant of this equation u will find that it is positive .

 

this implies that there will be two values of a

 

implies two triangles

Answer 2

The best way to solve this is to use the laws of sine and laws of cosine.

 

 

We can use law of sine when we know at least one angle and two sides, or at least one side and two angle measures.

 

Remember this:

 

Side a is opposite to angle A.

Side b is opposite to angle B.

Side c is opposite to angle C.

 

 

Law of sine is defined by

 

a / sin(A) = b / sin(B)

 

 

Law of cosine is defined by

 

c² = a² + b² -2abCos(C)

 

 

Okay, now that you know how to use these laws, lets apply them.

 

We are given angle B and side b.  Use law of sine to find Angle C because we know side c.

 

 

28 / Sin(84) = 25 / Sin(C)

 

Once you have C, you add the values of C and B.  Subtract their sum from 180 to find the third angle.  Three angles gets us a triangle.

 

Use the other laws to find the other two missing sides.

 

 

If we have one triangle, then we can draw line segments inside the triangle to obtain more triangles. 

 

But these measurements alone is enough to determine one triangle.