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The problem asks to find the dot product of vectors \( \overrightarrow{BC} \) and \( \overrightarrow{AC} \), where the geometric diagram represents a parallelogram with the following points:

- \( A(0, 0) \), assuming the origin.
- \( B(10, 0) \), since the length \( AB = 10 \).
- \( C(10, 7) \), as \( BC \) is 7 units long and it rises vertically from \( B \).
- \( D(x, y) \), which we can ignore for now since we are focused on \( BC \) and \( AC \).

### Step 1: Coordinates of Points
- \( A(0, 0) \)
- \( B(10, 0) \)
- \( C(10, 7) \)

### Step 2: Find the Vectors
- Vector \( \overrightarrow{BC} = C - B = (10, 7) - (10, 0) = (0, 7) \)
- Vector \( \overrightarrow{AC} = C - A = (10, 7) - (0, 0) = (10, 7) \)

### Step 3: Dot Product Formula
The dot product of two vectors \( \overrightarrow{u} = (u_1, u_2) \) and \( \overrightarrow{v} = (v_1, v_2) \) is given by:

\[
\overrightarrow{u} \cdot \overrightarrow{v} = u_1 v_1 + u_2 v_2
\]

For \( \overrightarrow{BC} = (0, 7) \) and \( \overrightarrow{AC} = (10, 7) \):

\[
\overrightarrow{BC} \cdot \overrightarrow{AC} = (0)(10) + (7)(7) = 0 + 49 = 49
\]

Thus, the dot product \( \overrightarrow{BC} \cdot \overrightarrow{AC} \) is 49.

The closest answer to this in the options given is not explicitly there, but it seems there's an interpretation mistake (with fractions) or another component. Let me know if you need further clarification or details about the dot product and vector operations.

Do you have any more questions? Here are 5 related questions:

1. What is the physical interpretation of the dot product in geometry?
2. How does the angle between two vectors affect the dot product?
3. What is the formula for the dot product in three dimensions?
4. Can the dot product be negative? If so, under what conditions?
5. What is the difference between the dot product and the cross product?

### Tip: Always remember that the dot product gives a scalar value and involves both magnitude and direction of vectors!

Find the dot product of vectors BC and AC based on the diagram of a parallelogram with points A(0,0), B(10,0), and C(10,7).

This problem involves finding the dot product of vectors BC and AC in a parallelogram. The points A(0,0), B(10,0), and C(10,7) are used to calculate the vectors and apply the dot product formula. Suitable for students in Grades 10-12.

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