Graph the following points on the coordinate plane. Find the measure of ∠
D to the nearest degree.

D (4, 3), E (4, -2), F (-3, -2)

To solve this problem, we'll follow these steps:

1. **Graph the points \(D(4, 3)\), \(E(4, -2)\), and \(F(-3, -2)\) on the coordinate plane**.
2. **Determine the vectors corresponding to the sides of the angle \( \angle D \)**.
3. **Calculate the angle \( \angle D \) formed between these vectors using the dot product formula**.
4. **Convert the angle to degrees and round to the nearest degree**.

### Step 1: Graph the Points

The coordinates are:
- \( D(4, 3) \)
- \( E(4, -2) \)
- \( F(-3, -2) \)

### Step 2: Calculate the Vectors

The vectors from \(D\) to \(E\) and \(D\) to \(F\) are calculated as:

\[
\overrightarrow{DE} = E - D = (4 - 4, -2 - 3) = (0, -5)
\]
\[
\overrightarrow{DF} = F - D = (-3 - 4, -2 - 3) = (-7, -5)
\]

### Step 3: Calculate the Angle Using the Dot Product Formula

The dot product of two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \) is given by:

\[
\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2
\]

For the vectors \( \overrightarrow{DE} \) and \( \overrightarrow{DF} \):

\[
\overrightarrow{DE} \cdot \overrightarrow{DF} = (0)(-7) + (-5)(-5) = 0 + 25 = 25
\]

The magnitude of the vectors is given by:

\[
|\overrightarrow{DE}| = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5
\]
\[
|\overrightarrow{DF}| = \sqrt{(-7)^2 + (-5)^2} = \sqrt{49 + 25} = \sqrt{74}
\]

Now, the cosine of the angle between the vectors is:

\[
\cos(\theta) = \frac{\overrightarrow{DE} \cdot \overrightarrow{DF}}{|\overrightarrow{DE}| |\overrightarrow{DF}|} = \frac{25}{5 \times \sqrt{74}} = \frac{25}{5\sqrt{74}} = \frac{5}{\sqrt{74}}
\]

Finally, the angle \( \theta \) is:

\[
\theta = \cos^{-1}\left(\frac{5}{\sqrt{74}}\right)
\]

### Step 4: Calculate the Angle and Convert to Degrees

Let's compute this angle using the cosine inverse function.The measure of \( \angle D \) is approximately **54 degrees**.

Would you like more details or have any other questions?

### Related Questions:
1. How do you find the length of a vector in 2D space?
2. What is the dot product of two vectors, and what does it represent?
3. How do you convert radians to degrees?
4. Can you explain the geometric interpretation of the dot product?
5. What would happen to the angle if one of the points was moved?
6. How do you determine if three points form a right triangle?
7. What is the significance of a negative dot product in determining angles?
8. How would this calculation change if the vectors were in 3D space?

**Tip:** When working with vectors and angles, always double-check your vector calculations, as a small mistake in direction can lead to an incorrect angle.

Learn how to graph points on a coordinate plane and find the measure of angle D using vector calculations and the dot product formula. This problem involves coordinate geometry concepts and is suitable for Grades 9-12 students.

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