Let's go over the questions from the image.

Question 8:

In the diagram, a rectangle $PQRS$ is given, with diagonals $PR$ and $QS$ intersecting at $M$. The lengths of segments $PM = 6x - 8$ and $QM = 2x + 4$ are provided. The question asks to determine the value of $x$.

Solution:

For a rectangle, the diagonals are equal in length and bisect each other, meaning $PM = QM$. So we can set up the equation:

$6x - 8 = 2x + 4$

Solve for $x$:

1. Subtract $2x$ from both sides:

$4x - 8 = 4$

2. Add 8 to both sides:

$4x = 12$

3. Divide by 4:

$x = 3$

Thus, the value of $x$ is $3$.

Would you like the details for the other questions as well, or do you have any specific queries? Here are five additional questions related to this:

1. How do we verify the equality of diagonals in a rectangle?
2. What is the significance of the intersection point in a rectangle's diagonals?
3. What are the properties of diagonals in other quadrilaterals, like a rhombus or square?
4. How would the problem change if this were a parallelogram instead of a rectangle?
5. Can the value of $x$ be negative in some problems like this?

Tip:

Always check if geometrical properties (like the equality of diagonals) hold before setting up your equation.