Question

the law of cosines for $\triangle rst$ can be set up as $5^2 = 7^2 + 3^2 - 2(7)(3)\cos(s)$. what could be true about \\(\triangle\\) law of cosines: $a^2 = b^2 + c^2 - 2bc\cos(a)$ \\(\bigcirc\\) $r = 5$ and $t = 7$ \\(\bigcirc\\) $r = 3$ and $t = 3$ \\(\bigcirc\\) $s = 7$ and $t = 5$ \\(\bigcirc\\) $s = 5$ and $t = 3$

Explanation:

Step1: Recall Law of Cosines Structure

The law of cosines is \(a^{2}=b^{2}+c^{2}-2bc\cos(A)\), where \(a\) is the side opposite angle \(A\), and \(b\) and \(c\) are the other two sides. In \(\triangle RST\), the formula is \(5^{2}=7^{2}+3^{2}-2(7)(3)\cos(S)\). So, the side opposite angle \(S\) is \(5\) (since \(a = 5\) when angle is \(A = S\)), and the other two sides are \(7\) and \(3\) (since \(b = 7\), \(c = 3\)). In triangle notation, sides are named opposite their respective angles: side \(r\) is opposite angle \(R\), side \(s\) is opposite angle \(S\), side \(t\) is opposite angle \(T\). So side \(s\) (opposite angle \(S\)) should be \(5\), and the other two sides (adjacent to angle \(S\)) are \(7\) and \(3\). So one of them could be \(t = 3\) (and the other \(r = 7\) or vice - versa, but looking at options:

Step2: Analyze Each Option

• Option 1: \(r = 5\) and \(t = 7\): Side opposite \(S\) is \(s\), not \(r\), so wrong.
• Option 2: \(r = 3\) and \(t = 3\): The two sides adjacent to \(S\) are \(7\) and \(3\), not both \(3\), wrong.
• Option 3: \(s = 7\) and \(t = 5\): Side opposite \(S\) should be \(5\), not \(7\), wrong.
• Option 4: \(s = 5\) (opposite angle \(S\)) and \(t = 3\) (one of the other sides), which matches the law of cosines formula.

Answer:

\(s = 5\) and \(t = 3\) (the fourth option: \(s = 5\) and \(t = 3\))