https://file.aiquickdraw.com/mathbot/file/1728498167226vxluffoy.jpg

To solve the problem shown in the image, we need to calculate the distance between the two points \( Q(-1, -6) \) and \( E(-9, 2) \) in the coordinate plane.

The formula for the distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Identify the coordinates
- Point \( Q \) has coordinates \( (-1, -6) \), so \( x_1 = -1 \) and \( y_1 = -6 \).
- Point \( E \) has coordinates \( (-9, 2) \), so \( x_2 = -9 \) and \( y_2 = 2 \).

### Step 2: Apply the distance formula
Substitute the values into the formula:

\[
d = \sqrt{((-9) - (-1))^2 + (2 - (-6))^2}
\]

\[
d = \sqrt{(-9 + 1)^2 + (2 + 6)^2}
\]

\[
d = \sqrt{(-8)^2 + (8)^2}
\]

\[
d = \sqrt{64 + 64}
\]

\[
d = \sqrt{128}
\]

\[
d = 8\sqrt{2}
\]

Thus, the exact distance between the points \( Q \) and \( E \) is \( 8\sqrt{2} \).

Let me know if you want further details or have any questions.

Here are 5 related questions for practice:
1. What is the distance between \( (3, 4) \) and \( (7, 1) \)?
2. How do you derive the distance formula from the Pythagorean theorem?
3. What is the midpoint between \( Q(-1, -6) \) and \( E(-9, 2) \)?
4. If the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) have the same x-coordinate, how would the distance formula simplify?
5. How do you interpret the result \( 8\sqrt{2} \) geometrically?

**Tip**: Always square the differences in coordinates carefully to avoid sign errors when applying the distance formula!

Calculate the distance between the points Q = (-1, -6) and E = (-9, 2) in the coordinate plane. Give an exact answer (not a decimal approximation).

Find the exact distance between points Q(-1, -6) and E(-9, 2) using the distance formula in coordinate geometry. A step-by-step solution is provided, explaining the application of the Pythagorean theorem and the distance formula. Suitable for high school students.

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