1. **Problem statement:** Points P and Q are on opposite sides of a lake, R is on land, M is the midpoint of RP, and N is the midpoint of RQ. The segment MN is measured as $(6k + 8)$ meters. We need to find the width of the lake, which is the distance $PQ$. 2. **Key formula and concept:** Since M and N are midpoints of segments $RP$ and $RQ$ respectively, by the Midsegment Theorem in triangles, segment $MN$ is parallel to $PQ$ and its length is half the length of $PQ$. Mathematically, $$MN = \frac{1}{2} PQ.$$ This means $$PQ = 2 \times MN.$$ 3. **Part (a) - Expressing $PQ$ in terms of $k$:** Given $MN = 6k + 8$, substitute into the formula: $$PQ = 2 \times (6k + 8) = 12k + 16.$$ So, the width of the lake $PQ$ is $12k + 16$ meters. 4. **Part (b) - Checking the possibility of $MN = 40$ m and $PQ = 70$ m:** Using the Midsegment Theorem, $PQ$ should be twice $MN$, so: $$PQ = 2 \times MN = 2 \times 40 = 80.$$ But the surveyor says $PQ = 70$ m, which is not equal to $80$ m. Therefore, it is **not possible** for $MN = 40$ m and $PQ = 70$ m simultaneously if M and N are midpoints as described, because $PQ$ must be exactly twice $MN$. **Final answers:** - (a) $PQ = 12k + 16$ meters. - (b) No, it is not possible for $MN = 40$ m and $PQ = 70$ m under the given conditions.