Question

Please help me figure how to explain ??

Answer 1

First of all, let's explain what SSA means. This is a concept of triangles and means Side, Side, Angle. This happens when we know two sides and an angle that is not the angle between the sides as shown in the Figure below.

On the other hand, the law of sines states that for the triangle below it is true that:


`(a)/(sinA)= (b)/(sinB)= (c)/(sinC)`

According to the problem we will explain three cases that result in zero, one or two triangles while applying SSA. Let's solve this problem with examples.

First. Case zero solution (No solution)

Suppose that:


`b=8 \\ c=9 \\ B=75^(\circ)`

So let's start by finding angle C. According to the law of sines:


`(c)/(sinC)= (b)/(sinB) \\ \\ \therefore (9)/(sinC)= (8)/(sin75^(\circ)) \\ \\ \therefore sinC= (9sin75^(\circ))/(8) \\ \\ \therefore C=sin^(-1)(1.086)`

If you try to take the arcsin of a number that's larger than one it is not going to work. Therefore there is no solution in this case.

Second. Case one solution

Suppose that:


`b=10 \\ c=9 \\ B=42^(\circ)`

Then according to the law of sines:


`(b)/(sinB)= (c)/(sinC) \\ \\ \therefore (10)/(sin42^(\circ))= (9)/(sinC) \\ \\ \therefore sinC= (9sin42^(\circ))/(10)=0.602 \\ \\ \therefore C=sin^(-1)(0.602)=37.01^(\circ) \ or \ C=180^(\circ)-37.01^(\circ)=142.99^(\circ)`

But we know that:


`A+B+C=180^(\circ) \\ \\ If \ C=37.01^(\circ) \ then: \\ A=180^(\circ)-42^(\circ)-37.01^(\circ)=79.01^(\circ) \\ It's \ a \ solution \\ \\ If \ C=142.99^(\circ) \ then: \\ A=180^(\circ)-42^(\circ)-142.99^(\circ)=-4.99^(\circ) \\ It's \ not \ a \ solution`

Therefore we have one solution

Third. Two solutions.

Suppose that:


`B=30^(\circ) \\ b=7 \\ c=8`

So:


`(b)/(sinB)= (c)/(sinC) \\ \\ \therefore (7)/(sin30^(\circ))=(8)/(sinC) \\ \\ \therefore sinC=(8sin30^(\circ))/(7)=(4)/(7) \\ \\ \therefore C=sin^(-1)((4)/(7))=34.85^(\circ) \ or \ C=180^(\circ)-34.85^(\circ)=145.15^(\circ) \\ \\ Thus: \\ \\ If \ C=34.85^(\circ) \ then:\\ A=180^(\circ)-30^(\circ)-34.85^(\circ)=115.15^(\circ) \\ It's \ a \ solution \\ \\ If \ C=145.15^(\circ) \ then:\\ A=180^(\circ)-30^(\circ)-145.15^(\circ)=4.85^(\circ) \\ It's \ a \ solution`

Therefore there are two solutions.

Conclusion: For triangles to make sense the shortest side must be across the lowest angle and the longest side must be across the largest angle.