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The image you've provided shows a rhombus with vertices \( P \), \( Q \), \( R \), and \( S \). You are asked to find the measure of the indicated side \( PS \), and you are given that the diagonal \( RS \) is 12 units long. In a rhombus, all sides are equal, and the diagonals bisect each other at right angles.

### Step-by-step solution:

1. In a rhombus, the diagonals bisect each other perpendicularly (at 90 degrees) and divide the rhombus into four congruent right triangles.
2. The diagonal \( RS = 12 \). Since diagonals bisect each other, half of \( RS \) is \( \frac{12}{2} = 6 \). Therefore, \( RS \) is split into two segments of 6 units each.
3. The diagonals in a rhombus are also perpendicular to each other, so triangle \( PRS \) is a right triangle.
4. In triangle \( PRS \), \( PR = PQ = PS = QR \) (since all sides of a rhombus are equal), and you can use the Pythagorean theorem to find \( PS \).
5. Let the length of each side (including \( PS \)) be \( x \). Applying the Pythagorean theorem to the right triangle \( PRS \):

\[
x^2 = \left(\frac{RS}{2}\right)^2 + \left(\frac{PQ}{2}\right)^2
\]

\[
x^2 = 6^2 + 6^2
\]

\[
x^2 = 36 + 36 = 72
\]

\[
x = \sqrt{72} = 6\sqrt{2}
\]

Thus, the length of side \( PS \) is \( 6\sqrt{2} \) units.

Would you like further details or have any questions about the explanation?

Here are some related questions to expand the concept:
1. What is the formula for the area of a rhombus using its diagonals?
2. How do you find the length of the diagonals if only the side lengths of a rhombus are given?
3. What is the relationship between the angles formed by the diagonals in a rhombus?
4. How do you calculate the perimeter of a rhombus given one diagonal?
5. Can a square be considered a special type of rhombus? Why?

**Tip**: In any rhombus, the diagonals always bisect each other at right angles, which helps in applying the Pythagorean theorem for finding side lengths.

Find the measure of the indicated side PS in the given rhombus, where RS = 12 units.

Learn how to find the length of a side in a rhombus where the diagonal is given, using the Pythagorean theorem. This problem explores key geometry concepts such as rhombus properties and the relationship between diagonals and side lengths. Suitable for students in Grades 9-10.

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