The problem is asking to prove that the diagonals of a rectangle bisect each other. Here's a breakdown of how we can approach the proof:

Given:

• A rectangle $ABCD$.
• Diagonals $AC$ and $BD$ intersect at point $M$.

We need to prove that $M$ is the midpoint of both diagonals $AC$ and $BD$, meaning that: $AM = MC \quad \text{and} \quad BM = MD.$

Vector Approach:

We are provided with the hint that suggests using vectors:

• $\overrightarrow{AB} = \vec{a}$
• $\overrightarrow{AD} = \vec{b}$

Now, consider the vector for the diagonal $AC$ and $BD$. We can express the vectors for points $A$, $B$, $C$, and $D$ using $\vec{a}$ and $\vec{b}$.

Step-by-Step Solution:

1. Express $M$ in terms of vectors:

   • The coordinates of $M$, the midpoint, will be the average of the vectors representing $A$ and $C$, and similarly for $B$ and $D$.

   Let $A = \vec{0}$ (origin), $B = \vec{a}$, $D = \vec{b}$, and $C = \vec{a} + \vec{b}$.

2. Vector for midpoint $M$ of diagonal $AC$: $M = \frac{A + C}{2} = \frac{\vec{0} + (\vec{a} + \vec{b})}{2} = \frac{\vec{a} + \vec{b}}{2}$

3. Vector for midpoint $M$ of diagonal $BD$: Similarly, for diagonal $BD$, we calculate the midpoint as: $M = \frac{B + D}{2} = \frac{\vec{a} + \vec{b}}{2}$

Thus, $M$ is the same point for both diagonals, meaning that the diagonals bisect each other.

Conclusion:

The diagonals of a rectangle bisect each other at the midpoint $M$. This is shown by calculating the midpoint for both diagonals using vector addition and confirming that the result is the same for both diagonals.

Would you like any additional details or have any questions? Here are some relative questions for further exploration:

1. How would the proof change if the figure was a parallelogram instead of a rectangle?
2. Can the midpoint concept be extended to other polygons like rhombuses or trapezoids?
3. Why is vector addition a useful tool in geometry proofs like this one?
4. How does the symmetry of a rectangle influence the properties of its diagonals?
5. What would change if the rectangle was not oriented with sides parallel to the axes?

Tip:

In vector geometry, always start by assigning vector coordinates to key points, which simplifies the proof process through addition and midpoint calculations.