To solve for the length $RU$ in the given right triangle, we will apply the Pythagorean Theorem, which is useful when working with right triangles. According to the diagram:

1. $\triangle RUS$ is a right triangle, with a right angle at $U$.
2. The hypotenuse $RS$ is 65 units.
3. One leg, $SU$, is unknown.
4. The leg $RU$ is what we need to find.

First, observe that the full base $RT$ is given as 84, and the hypotenuse $ST$ of the larger triangle is 105. But we will focus on the right triangle $RUS$ for now.

By the Pythagorean Theorem: $RS^2 = RU^2 + SU^2$ Where:

• $RS = 65$
• $SU$ is unknown.

We cannot directly solve this without additional information about $SU$. However, we know that $UT$ can be derived because $UT$ is also a leg of the right triangle $\triangle SUT$ (since $U$ is a right angle). Using the Pythagorean theorem for $\triangle SUT$: $ST^2 = SU^2 + UT^2$ $105^2 = SU^2 + 84^2$ $11025 = SU^2 + 7056$ $SU^2 = 11025 - 7056 = 3969$ $SU = \sqrt{3969} = 63$

Now that we know $SU = 63$, we can go back to the triangle $RUS$ and solve for $RU$: $65^2 = RU^2 + 63^2$ $4225 = RU^2 + 3969$ $RU^2 = 4225 - 3969 = 256$ $RU = \sqrt{256} = 16$

Thus, the length of $RU$ is 16 units.

Let me know if you'd like further details or have any questions! Here are some related questions:

1. How do we find the length of $UT$?
2. What is the relationship between the Pythagorean theorem and right triangles?
3. How can the Pythagorean theorem be applied in three-dimensional geometry?
4. How does the cosine function relate to finding unknown sides in a triangle?
5. How would we solve for angles in this triangle using trigonometry?

Tip: Always verify which triangle and which sides you are working with, especially when multiple triangles share sides or angles.