Sunday, May 8, 2011

Sunday, May 8, 2011

Some of these problems are based on triangle inequality.

Draw triangle whose sides are of length 4,5 and 7 cms. Try doing the same exercise with sides of length 3,4, 8 cms. Why can't you make a triangle in the 2nd case?

Show that in any triangle, the longest side is always smaller than the sum of other two sides.

Given three positive numbers a, b, c in a non-decreasing order, such that c is at most a+b, then you can always form a triangle with side length a, b and c.

Show tha no side of a triangle has a length larger than half of it's perimeter.

Show that in a convex quadrilateral (no interior angle bigger than 180 degrees), the sum of the length of two opposite sides is no larger than the sum of the lengths of diagonals.

Where to place the Strandherd-Armstrong bridge?
Suppose that Rideau River is a straight line, and two friends live on opposite side of the river. Where should the new bridge be placed so that their travel time is as small as possible. What if, there are two friends on one side, and one on another, what if two and two, and what if two communities?
what if the city is rich enough to approve two bridges?

The distance between big apple on Hwy 401 to Montreal is 400 kms, and to Toronto is 160 Kms.
Does that mean that the distance between Montreal and Toronto is 260 kms? Also, the distance to Ottawa is about 350kms. Does that mean that Montreal to Ottawa is 50kms. In general, if you are given distances from a point to some of the cities, what can you tell about minimum and maximum distances between these cities.

What will happen if we have the following variation - Sum of the two sides is always smaller than the third, i.e, To go from a to b, it is always best to go via c! In other words, it's always best to take a detour. Can we ever reach b from a?