Dr. Huasong Yin, Author of MathCounts Preparation: How to Excel at Middle School Math Competitions
Question of the Day 3/7/2014
Question: How many pairs of perpendicularly intersecting line segments can be
drawn using the points of the following 3 by 3 grid as endpoints? Segments must
intersect at one point, either at an endpoint or within the segment.
Dr. Huasong Yin, Author of MathCounts Preparation: How to Excel at Middle School Math Competitions
Solution:
49+21+8=78.
Horizontal and vertical line segments: Any two
perpendicularly intersecting line segments will span a
rectangle. We will count by the type of rectangles
contained in the graph.
Case 1: there are 4 unit squares (the 4 smallest ones).
Each contains 4 pairs of perpendicular line segments
(The intersection points are the 4 corners). 4*4=16
Case 2: there are 4 rectangles of area 2. Each contains 6
pairs of perpendicular line segments that span it. (The 6
grid dots on it as the intersection points). 4*6=24
Case 3: there is only one rectangle of area 4, the big
square. It contains 9 pairs of perpendicular line
segments that span it. (The 9 dots in the graph as the
intersection point).
16+24+9=49.
Diagonals of slope 1 and -1:
There are 4 pairs containing all red diagonals.
There are only two pairs of perpendicular lines
containing exactly one red diagonal. 2*4=8.
There are 4 pairs containing all black diagonals of
shorter length.
There are 4 pairs of all black diagonals, one shorter
length and the other longer length.
There is only 1 pair of all black diagonals, both of longer
length.
8+4+4+4+1=21.
line segments of slopes 2, 1/2, -2, -1/2 in the grid:
Line segment of slope 2 is perpendicular to line segment
of slope -1/2. As shown in the graph, there are 4 such
pairs.
Symmetrically there are 4 pairs of perpendicular line
segments with slopes 1/2 and -2.
4+4= 8.
Answer: 78