https://file.aiquickdraw.com/mathbot/file/1728224888515lsek2yph.jpg

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 \( 3 \).

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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.

A rectangle PQRS is given with diagonals PR and QS intersecting at M. The lengths of PM = 6x - 8 and QM = 2x + 4 are given. Find the value of x.

Learn how to solve a problem involving the equality of diagonals in a rectangle, where PM = 6x - 8 and QM = 2x + 4. We solve for x using algebra and properties of rectangles. Suitable for Grade 8-9 students.

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