The question asks to find the length $RU$ in the triangle diagram provided. Here's how to solve it step by step:

Observations:

1. Given values:

   • $RS = 13$ (leg of the smaller triangle),
   • $ST = 20$ (hypotenuse of the larger triangle),
   • $RT = 16$ (base of the larger triangle).

2. The problem involves two right triangles:

   • Small triangle $\triangle RSU$: $RS$ is one leg, $SU$ is the other leg, and $RU$ is the hypotenuse.
   • Large triangle $\triangle RST$: $RT$ is the base, $ST$ is the hypotenuse.

3. We use the Pythagorean Theorem to find the missing sides.

Step 1: Find $RU$ using the smaller triangle

Using the Pythagorean Theorem for $\triangle RSU$: $RU^2 = RS^2 + SU^2$

To calculate $SU^2$, note that $SU$ is a common side to both triangles. First, we'll use the larger triangle to calculate $SU$.

Step 2: Find $SU$ using the larger triangle

Using the Pythagorean Theorem for $\triangle RST$: $ST^2 = RT^2 + SU^2$

Substitute the given values: $20^2 = 16^2 + SU^2$ $400 = 256 + SU^2$ $SU^2 = 400 - 256 = 144$ $SU = \sqrt{144} = 12$

Step 3: Substitute $SU = 12$ into the equation for $RU$

Now, go back to $\triangle RSU$ and substitute $RS = 13$ and $SU = 12$: $RU^2 = RS^2 + SU^2$ $RU^2 = 13^2 + 12^2$ $RU^2 = 169 + 144 = 313$ $RU = \sqrt{313} \approx 17.7$

Rounding to the nearest whole number: $RU \approx 18$

Final Answer:

The length of $RU$ is approximately 18.

Would you like additional details on the steps or concepts used?

Here are 5 related questions to explore further:

1. What is the Pythagorean Theorem, and how is it used in right triangles?
2. How do you calculate the hypotenuse when given two legs of a triangle?
3. How can you confirm if a triangle is a right triangle using its side lengths?
4. What are real-world applications of the Pythagorean Theorem?
5. How can trigonometric ratios be used to find unknown sides of triangles?

Tip: Always double-check calculations involving square roots to avoid small errors.