The diagram represents a right triangle $\triangle LKN$, where $\overline{LM}$ is the hypotenuse and $\overline{KN}$ is the side we need to find. We are given the following:

• $\overline{LN} = 29$
• $\overline{NM} = 48$
• $\overline{LM} = 52$

Since $\triangle LKN$ is a right triangle, we can use the Pythagorean theorem to find $\overline{KN}$. The theorem states:

$\overline{LM}^2 = \overline{LN}^2 + \overline{KN}^2$

Step 1: Rearrange to solve for $\overline{KN}^2$:

$\overline{KN}^2 = \overline{LM}^2 - \overline{LN}^2$

Step 2: Substitute the given values:

$\overline{KN}^2 = 52^2 - 29^2$

Step 3: Perform the calculations:

$\overline{KN}^2 = 2704 - 841 = 1863$

Step 4: Take the square root to find $\overline{KN}$:

$\overline{KN} = \sqrt{1863} \approx 43.2$

Step 5: Round to the nearest whole number:

$\overline{KN} \approx 43$

Final Answer:

The length of $\overline{KN}$ is approximately 43 units.

Do you have any questions, or would you like additional details about this solution? Here are some related questions to explore:

1. What is the Pythagorean theorem, and how is it used to solve problems like this?
2. How can we verify if a triangle is a right triangle using side lengths?
3. How can we find the missing side in a triangle when the hypotenuse and one leg are known?
4. What are some practical applications of the Pythagorean theorem in real life?
5. How do we compute square roots efficiently for larger numbers?

Tip: When working with approximate square roots, double-check calculations to ensure the correct level of precision!