The exercises in the image involve the following tasks:

Exercise 1

Determine the distance between the given pairs of points in the plane:

1. $P(5,7)$ and $Q(-1,3)$
2. $A(-4,-2)$ and $B(1,5)$
3. $C(0,3)$ and $D(4,0)$
4. $E(2,2)$ and $F(5,6)$

To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Exercise 2

Find the perimeter of the quadrilateral with vertices:

• $A(-4,-2)$
• $B(1,5)$
• $C(2,2)$
• $D(-4,0)$

To solve this, calculate the distances between each pair of consecutive vertices (AB, BC, CD, DA) and sum them to obtain the perimeter.

Exercise 3

Determine the lengths of the sides of the triangle with vertices:

• $P(0,4)$
• $Q(2,3)$
• $T(-2,0)$

Then, plot the triangle. The distance formula can be used for each pair of points (PQ, QT, and TP).

Would you like detailed calculations for each question, or is there a specific part you’d like to focus on?

Here are some related questions that can expand on the concepts:

1. How do you calculate the midpoint between two points?
2. What is the formula for calculating the area of a triangle given three vertices?
3. How can you verify if three points are collinear?
4. What is the formula for the perimeter of different polygons based on their vertices?
5. How can the distance formula be applied in three-dimensional space?

Tip: When working with distances, always double-check your calculations to avoid errors, especially with signs in coordinates.