Given a parallelogram $ABCD$, the sides and angles provided give us enough information to compute the required lengths. Let's break down the information and the steps:

1. Given Data:

• $AB = 3\sqrt{2}$, $AD = 8$, and $\angle BAD = 45^\circ$.
• $M$ is a point on $CD$, and $CM : MD = 1 : 2$.
• $N$ is a point on $AD$, and $AN : ND = 3 : 1$.

We are tasked with finding the lengths of:

1. $AM$
2. $BN$

Step 1: Analyze the Parallelogram

In any parallelogram:

• Opposite sides are equal: $AB = CD$, $AD = BC$.
• Opposite angles are equal: $\angle BAD = \angle BCD$, and adjacent angles are supplementary.
• Vectors can be helpful in breaking down the problem geometrically.

Step 2: Work on Length of $AM$

Let’s use vector coordinates. Assign coordinates for the vertices:

• $A = (0, 0)$,
• $B = (8, 0)$ (since $AD = 8$),
• $D = (0, 8)$ (perpendicular from $A$),
• $C = (8, 8)$.

Now, since $CM : MD = 1 : 2$, the point $M$ divides $CD$ in the ratio $1:2$. Hence, the coordinates of $M$ can be found using the section formula: $M = \left( \frac{2(8) + 1(8)}{1+2}, \frac{2(8) + 1(0)}{1+2} \right) = \left( 8, \frac{16}{3} \right).$

The length of $AM$ is the distance between $A(0, 0)$ and $M(8, \frac{16}{3})$. Using the distance formula: $AM = \sqrt{(8-0)^2 + \left( \frac{16}{3} - 0 \right)^2} = \sqrt{64 + \left( \frac{16}{3} \right)^2} = \sqrt{64 + \frac{256}{9}}.$ Simplify: $AM = \sqrt{\frac{576}{9} + \frac{256}{9}} = \sqrt{\frac{832}{9}} = \frac{\sqrt{832}}{3} = \frac{4\sqrt{52}}{3}.$ Hence, $AM = \frac{8\sqrt{13}}{3}$.

Step 3: Work on Length of $BN$

Since $N$ divides $AD$ in the ratio $AN : ND = 3:1$, the coordinates of $N$ can be found as: $N = \left( \frac{3(0) + 1(0)}{3+1}, \frac{3(8) + 1(0)}{3+1} \right) = (0, 6).$

The length of $BN$ is the distance between $B(8, 0)$ and $N(0, 6)$. Using the distance formula: $BN = \sqrt{(8-0)^2 + (0-6)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10.$

Final Answers:

• $AM = \frac{8\sqrt{13}}{3}$,
• $BN = 10$.

Would you like further explanation or details on any step?

Here are some related questions:

1. How do we calculate section points using the section formula?
2. What is the geometrical significance of ratios in division of a line?
3. How can vectors be used to find distances in geometry problems?
4. What are the properties of parallelograms that are useful in solving geometric problems?
5. How do we find distances between points in the coordinate plane?

Tip: When dividing a line segment in a given ratio, the section formula simplifies computations in geometry.