(Originais Teachy 2023) - Question Medium of Mathematics

In a school, a survey was conducted on students who practiced sports. It was found that 60% of the students practiced soccer, 30% practiced volleyball, and 20% practiced both sports. If a student is randomly selected, what is the probability that he practices soccer, given that he practices volleyball?

In a civil engineering project, an architect needs to divide a rectangular plot of land into two equal parts for the construction of two semi-detached houses. The coordinates of the vertices of the land on the Cartesian plane are A(0, 0), B(30, 0), C(30, 20), and D(0, 20). Using concepts of analytic geometry, determine the coordinates of the point that represents the midpoint of side CD of the land, which will be the dividing point between the two new properties. Then, verify if the obtained midpoint point is indeed on segment CD, ensuring the accuracy of the project. Explain the reasoning used to determine the coordinates and to verify the positioning of the midpoint on the land.

Consider a Cartesian coordinate system where a point P has coordinates (a, b) and a point Q has coordinates (c, d). The absolute value of a number is defined as its distance from the origin of this system. If point P is in the second quadrant and point Q is in the fourth quadrant, with a, b, c, and d being integers, and it is known that the absolute difference between the y coordinates of P and Q is 15, determine the value of a - c, knowing that a < 0 and c > 0.