3. Construct a triangle \( L M N \), such that \( L M=4 \mathrm{~cm} \) \( M N=5 \mathrm{~cm} \) and \( L N=3 \mathrm{~cm} \).

Sure! Here's a step-by-step guide to constructing triangle \( LMN \) with sides \( LM = 4 \) cm, \( MN = 5 \) cm, and \( LN = 3 \) cm using a compass and straightedge. ### **Materials Needed:** - Ruler - Compass - Pencil - Drawing paper ### **Construction Steps:** 1. **Draw the Base \( LM \):** - **Step 1:** Using the ruler, draw a straight horizontal line segment and label its endpoints as \( L \) and \( M \). - **Step 2:** Ensure that the length of \( LM \) is exactly **4 cm**.  2. **Construct a Circle with Center \( L \) and Radius \( LN = 3 \) cm:** - **Step 3:** Place the compass point on point \( L \). - **Step 4:** Adjust the compass to a width of **3 cm**. - **Step 5:** Draw a circle centered at \( L \). This circle represents all possible locations for point \( N \) that are **3 cm** away from \( L \).  3. **Construct a Circle with Center \( M \) and Radius \( MN = 5 \) cm:** - **Step 6:** Without changing the compass width, adjust it to **5 cm**. - **Step 7:** Place the compass point on point \( M \). - **Step 8:** Draw a second circle centered at \( M \). This circle represents all possible locations for point \( N \) that are **5 cm** away from \( M \).  4. **Find the Intersection Point \( N \):** - **Step 9:** Observe where the two circles intersect. There will be two possible intersection points above and below the line \( LM \). - **Step 10:** Choose one of the intersection points and label it as \( N \).  5. **Draw the Triangle \( LMN \):** - **Step 11:** Using the ruler, draw straight lines connecting \( L \) to \( N \) and \( M \) to \( N \). - **Step 12:** Verify the lengths: - \( LN \) should be **3 cm**. - \( MN \) should be **5 cm**.  ### **Final Triangle:** You should now have triangle \( LMN \) with the specified side lengths: - \( LM = 4 \) cm - \( MN = 5 \) cm - \( LN = 3 \) cm  ### **Visualization:** While I can't provide actual images here, the descriptions above correspond to the typical steps used in geometric constructions. Each step builds upon the previous one to ensure accuracy in creating the desired triangle. **Tips:** - **Precision is key:** Make sure your compass settings are exact to ensure the sides are the correct lengths. - **Label clearly:** Clearly labeling points helps in avoiding confusion, especially when dealing with multiple intersection points. - **Use light pencil strokes:** This allows for easy corrections if needed. If you follow these steps carefully, you should successfully construct the desired triangle \( LMN \) with the given side lengths.