#
SAT Question of the Day June 15 2013

### Four distinct lines lie in a plane, and exactly two of them are parallel. Which of the following could be the number of points where at least two of the lines intersect?

### . Three

### . Four

### . Five

### A. only

### B. only

### C. and only

### D. and only

### E. , , and

### In a math question concentrated on Geometry I recommend to make a little diagram.

### We have 4 lines in the plane. Exactly two of them are parallel – implying they never cross.

### Now lets call these lines A and B and lets draw them parallel to the bottom of the page.

### ————————————————————————

### ————————————————————————

### The remaining two lines (C and D) are not parallel to one another or to A or B.

### This mean we can imagine they are forming a gigantic

# \ /

# \ /

# \ /

# X

# / \

# / \

# / \

### Now lets move this X up.

### It is clear C and D meet exactly once. C intersect A once and intersect B once. The same is true for D.

### We have an option of 1+2+2 intersections.

### However, if we continue to move the X up and the intersection between C and D is now overlapping one of the parallel lines we reduced two points.

### We conclude there are either 5 or 3 points were the lines intersect.

### 30% out of 218,000 got it right – a surprisingly difficult question.

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