Question: 11. Seven squares of unit length are aligned edge-to-edge as shown. A straight line drawn from $P$ to $Q$ makes a number of angles with the sides of the seven squares. What is the ratio $M? QN$, if the line $PQ$ is the diagonal of the rectangle formed by the seven squares.
What is the ratio $M Q : Q N$?
A 1 : 2
B 2 : 3
C 3 : 5
D 5 : 8
E 8 : 13
12. A cube has a dot at each vertex, at the midpoint of each edge and at the midpoint of each face.
How many straight lines pass through exactly 3 dots?
A 24
B 36
C 48
D 60
E 72
13. J.K. and M.N. are equilateral triangles with side-lengths $20$ and $26$ respectively, and the point $X$ on line segment $JK$ lies such that $QP$ and $MN$ have the same length and the same area. The segment $QP$ is perpendicular to the line segment $JN$. What is the value of $n$?
What is the value of $n$?
A 24
B 23
C 22
D 21
E 20
1. **Problem 11:** Seven unit squares are aligned edge-to-edge forming a rectangle of dimensions $7 \times 1$. The diagonal $PQ$ of this rectangle is drawn. We need to find the ratio $MQ : QN$ where $M$ and $N$ are points on the diagonal related to the squares.
2. **Understanding the rectangle and diagonal:** The rectangle formed by $7$ unit squares has length $7$ and height $1$. The diagonal $PQ$ has length $$\sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}.$$
3. **Key insight:** The diagonal crosses the vertical lines between squares at points dividing the diagonal into segments. The ratio $MQ : QN$ corresponds to the ratio of segments along the diagonal related to the squares.
4. **Calculating the ratio:** The diagonal crosses the vertical edges at points dividing the length $7$ into $7$ equal parts. The vertical spacing is $1$ unit. The slope of the diagonal is $$\frac{1}{7}.$$
5. The ratio $MQ : QN$ corresponds to the ratio of horizontal distances along the diagonal. Since the diagonal crosses the vertical edges, the segments correspond to Fibonacci numbers in the ratio $5 : 8$.
6. **Answer for problem 11:** The ratio $MQ : QN = 5 : 8$, which corresponds to option D.
---
Since the user asked multiple questions but per instructions only the first is solved completely, we stop here.
**Final answer:** $5 : 8$ (Option D)