To solve the problem, we need to find the length of segment NP in triangle MNO, where MN = 6 units, MO = 4 units, and NO = 12 units. The angle bisector of angle M meets side NO at point P. We will use the Angle Bisector Theorem to find NP. ### Step-by-Step Explanation 1. **Understanding the Angle Bisector Theorem**: The Angle Bisector Theorem states that the ratio of the lengths of the two segments created by the angle bisector on the opposite side is equal to the ratio of the lengths of the other two sides of the triangle. In this case, if the angle bisector of angle M meets side NO at point P, then: \[ \frac{NP}{PO} = \frac{MN}{MO} \] 2. **Identifying the Lengths**: From the problem, we have: - MN = 6 units - MO = 4 units - NO = 12 units (which is the total length of segment NO) 3. **Setting Up the Ratio**: According to the Angle Bisector Theorem: \[ \frac{NP}{PO} = \frac{MN}{MO} = \frac{6}{4} = \frac{3}{2} \] This means that for every 3 parts of NP, there are 2 parts of PO. 4. **Expressing NP and PO in Terms of a Variable**: Let NP = 3x and PO = 2x. Since NP + PO = NO, we can write: \[ NP + PO = 3x + 2x = 5x \] Given that NO = 12 units, we can set up the equation: \[ 5x = 12 \] 5. **Solving for x**: To find x, we divide both sides by 5: \[ x = \frac{12}{5} = 2.4 \] 6. **Finding NP**: Now we can find NP: \[ NP = 3x = 3 \times 2.4 = 7.2 \text{ units} \] ### Conclusion Thus, the length of NP is **7.2 units**. ### Why Other Options Are Incorrect - **Option A (4.8 units)**: This value does not maintain the ratio established by the Angle Bisector Theorem. It would imply a different ratio of sides that does not correspond to the given lengths. - **Option C (8.0 units)**: This value also does not satisfy the ratio of 3:2 derived from the sides of the triangle. It would suggest that PO is too short compared to NP. - **Option D (18.0 units)**: This value is impossible since it exceeds the total length of NO (12 units). It violates the basic property of segment lengths in a triangle. ### Revision Summary - The Angle Bisector Theorem relates the lengths of segments created by an angle bisector to the lengths of the other two sides. - Set up a ratio based on the theorem and express the segments in terms of a variable. - Solve for the variable to find the lengths of the segments. - Always check that your calculated lengths do not exceed the total length of the side they are part of. By following these steps, you can effectively apply the Angle Bisector Theorem to similar problems in triangle geometry.