To answer this question, we can use trigonometric ratios. We have that the triangle is as follows:
As we can see we need to find the hypotenuse, and we can see that 5 is the opposite side to the angle 45.
We also know that:
[tex]\sin (\theta)=\frac{opposite}{hypotenuse}[/tex]
Therefore, we can use this trigonometric ratio to solve this question. Then we have:
[tex]\begin{gathered} \sin (45^{\circ})=\frac{5}{hypotenuse}=\frac{5}{KL} \\ \sin (45^{\circ})=\frac{5}{KL} \end{gathered}[/tex]
Now, we can multiply both sides by KL as follows:
[tex]\begin{gathered} KL\cdot\sin (45^{\circ})=KL\cdot\frac{5}{KL} \\ KL\cdot\sin (45^{\circ})=5 \\ KL\cdot\frac{\sin (45^{\circ})}{\sin (45^{\circ})}=\frac{5}{\sin (45^{\circ})} \\ KL=\frac{5}{\sin (45^{\circ})} \end{gathered}[/tex]
Notice that we also divided both sides by sin(45°). Then we have:
[tex]\begin{gathered} \sin (45^{\circ})=\frac{\sqrt[]{2}}{2} \\ KL=\frac{5}{\frac{\sqrt[]{2}}{2}}\Rightarrow KL=5\cdot\frac{2}{\sqrt[]{2}}=\frac{10}{\sqrt[]{2}}\cdot\frac{\sqrt[]{2}}{\sqrt[]{2}}=\frac{10\sqrt[]{2}}{2}=5\sqrt[]{2} \end{gathered}[/tex]
