How do you determine the number of possible triangles and find the measure of the three angles given a=8, b=10, mangleA=20?

1 Answer
Oct 31, 2017

A=20^@,B_1= 25^@19', C_1 = 134^@41' and

A=20^@,B_2= 154^@41'', C_2 = 5^@19'

Explanation:

Since the given information is for a SSA triangle it is the ambiguous case. In the ambiguous case we first find the height by using the formula h=bsin A.

Note that A is the given angle and its side is always a so the other side will be b .

So if A < 90^@ and if

  1. h < a < b then then there are two solutions or two triangles.

  2. h < b < a then there is one solution or one triangle.

  3. a < h < b then there is no solution or no triangle.

If A >=90^@ and if

  1. a > b then there is one solution or one triangle.

  2. a <=b there is no solution

Now let's use the Law of Cosine a^2 =b^2+c^2-2bc cos A and the

quadratic formula x=(-b+-sqrt(b^2-4ac)) /(2a)to figure out our solutions.

That is,

h=10sin20^@~~3.42, since 3.42 < 8 < 10 we have

h < a < b so we are looking for two solutions. Hence,

a^2 =b^2+c^2-2bc cos A

8^2=10^2 +c^2-2(10)(c) cos 20^@

64=100+c^2-(20cos20^@)c

0=c^2-(20cos20^@)c+36

c=((20cos20^@)+-sqrt((-20cos20^@)^2-4(1)(36) ))/2

c=((20cos20^@)+sqrt((-20cos20^@)^2-144 ))/2 or

c=((20cos20^@)-sqrt((-20cos20^@)^2-144 ))/2

:.c_1~~16.63 or c_2~~2.16

To find the measures of angle B we use the law of cosine and solve for B. That is,

B_1=cos^-1 [(8^2+c_1^2-10^2)/(2*c_1*8)]=25^@19'

and therefore

C_1=180^@-20^@-25^@ 19'=134^@41'

B_2=cos^-1 [(8^2+c_2^2-10^2)/(2*c_2*8)]=154^@41'

and therefore

C_2=180^@-20^@-154^@41'=5^@19'