Maths circle: Triangle inequality – We Solve Problems

Maths circle: Triangle inequality

Today we will solve some geometry problems using the triangle inequality. This is an inequality between the lengths of the sides of any triangle, or actually, between the distances of any three points. $\\$
For any two points, $A$ and $B$, the shortest path between them is a straight segment. Any other path will be longer. In particular, a path through another point, $C$, will be longer.
$$AC + BC \ge AB$$
The triangle inequality says that the sum of lengths of any two sides of a triangle is always larger than the length of the third side. The inequality only becomes an equality if $ABC$ is not actually a triangle and the point $C$ lies on the segment from $A$ to $B$. $\\$
Even though it is a really simple idea, it can be a really helpful tool in problem solving, as we will see today.

Problems:

Geometrical inequalities , Triangle inequality

The distance from school to the monument in the town centre is $4.2$ km, the distance from Anna’s house to school is $0.7$ km, the distance from Anna’s house to the monument is an integer number of kilometres. What exactly is it?

Geometrical inequalities , Triangle inequality

Tom and his grandma live at the same side of a straight river. Tom wants to visit his grandma, but also wants to stop by the river and fill his bottle with water. What is the shortest path that starts at his house, touches the river and ends at his grandma’s house?

Geometrical inequalities , Triangle inequality

A point $P$ is somewhere inside the triangle $\triangle ABC$. Show that $AP + BP < AC + BC$.

Geometrical inequalities , Triangle inequality

One side of a triangle has length $1$, the other has length $4$, and the third one has integer length. What is it?

Geometrical inequalities , Triangle inequality

There are 1450 km from London to Warsaw, Poland, and 680 km from Warsaw to Kyiv, Ukraine. The distance from London to New Delhi, India, is 6700 km and the distance from Kyiv to New Delhi is 4570 km. What is the distance from London to Kyiv?

Geometrical inequalities , Triangle inequality

Show that for any three points on the plane $A,B$ and $C$, $AB \ge |BC – AC|$.

Geometrical inequalities , Triangle inequality

Show that if all sides of a triangle have integer lengths and one of them is equal to $1$, then the other two have lengths equal to each other.

Geometrical inequalities , Triangle inequality

Two villages lie on opposite sides of a river whose banks are straight
lines. A bridge is to be built over the river perpendicular to the banks.
Where should the bridge be built so that the path from one village to
the other is as short as possible?

Geometrical inequalities , Triangle inequality

Quadrilateral $ABCD$ is completely inside a quadrilateral $EFGH$. Prove that the perimeter of $ABCD$ is smaller than the perimeter of $EFGH$.

Geometrical inequalities , Triangle inequality

A billiard ball lies on a table in the shape of an acute angle. How should
you hit the ball so that it returns to its starting location after hitting each of the two banks once? Is it always possible to do so? $\\$
(When the ball hits the bank, it bounces. The way it bounces is determined by the shortest path rule – if it begins at some point $A$ and ends at some point $B$ after bouncing, the path it takes is the shortest possible path that includes the bounce.)

Geometrical inequalities , Triangle inequality

There are $n$ mines and $n$ cities scattered across the land. Every mine has to have a rail connection to exactly one city. Railroads have to be straight and cannot cross other railroads. Is it always possible?

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