Question

find the measure of $overline{lm}$.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite to side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In \(\triangle KLM\), let \(a = 40\), \(b = 32\), and we want to find \(LM\) (let \(LM = c\)). First, we need to know the cosine of the included - angle. But if we assume this is a right - triangle (since no angle measure is given and we can use the Pythagorean theorem if it is a right - triangle), and check if it satisfies \(a^{2}+b^{2}=c^{2}\) or \(a^{2}+c^{2}=b^{2}\) or \(b^{2}+c^{2}=a^{2}\). Here, \(32^{2}+40^{2}=1024 + 1600=2624\). Let's assume the triangle is a right - triangle and by the Pythagorean theorem \(c=\sqrt{32^{2}+40^{2}}\) (assuming the sides \(32\) and \(40\) are the legs of a right - triangle).

Step2: Calculate the value of \(LM\)

\[

$$\begin{align*} LM&=\sqrt{32^{2}+40^{2}}\\ &=\sqrt{1024 + 1600}\\ &=\sqrt{2624}\\ &=\sqrt{16\times164}\\ & = 4\sqrt{164}\\ &=4\times2\sqrt{41}\\ &=8\sqrt{41}\approx 51.23 \end{align*}$$

\]

Answer:

\(8\sqrt{41}\)