Question

how many triangles exist that fit the following criteria? c = 135°, c = 7, a = 4

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{c}{\sin C}$. We know $C = 135^{\circ}$, $c = 7$, and $a = 4$. Substituting these values into the Law - of - Sines formula gives $\sin A=\frac{a\sin C}{c}$.
$\sin A=\frac{4\sin135^{\circ}}{7}$. Since $\sin135^{\circ}=\frac{\sqrt{2}}{2}$, then $\sin A=\frac{4\times\frac{\sqrt{2}}{2}}{7}=\frac{2\sqrt{2}}{7}\approx\frac{2\times1.414}{7}\approx0.404$.

Step2: Analyze the number of possible angles for A

Since $0^{\circ}Once we have determined angle $A$, we can find angle $B$ using the fact that the sum of the angles in a triangle is $180^{\circ}$ ($B = 180^{\circ}-A - C$), and then we can find side $b$ using the Law of Sines again. So, there is exactly 1 triangle that satisfies the given criteria.

Answer:

1