If a triangle has two sides with lengths 3 and 4, what should the length of the third side be in order to maximize the area of the triangle?

# Geometry

Puzzles involving any geometry, including shapes, areas, distances, and angles

Form four lines of four coins by moving exactly 3 of the coins below:

Rules:

- Coins cannot be stacked on top of one another.
- A line of five or more coins is still only considered one line.
- The centers of the coins must be precisely on the same straight line, but can be anywhere on the line.
- Lines can be any length, and can be horizontal, vertical, or diagonal.

## Inscribed Squares

Above are two identical isosceles right triangles containing two inscribed squares.

In one, a perfect square has been inscribed such that two sides line up with the two legs of the right triangle. In the other, a perfect square has been inscribed such that one side lines up with the hypotenuse of the right triangle.

Is the inscribed square on the left larger or the one on the right?

What is the length of the longest stick?

## Make Five into Four

Move exactly **3** matches to make these five squares into four squares of equal size, without any matches left over.

## Blue and Red Maze

In this maze, you must alternate going through blue and red doors – in other words, you cannot go through two blue doors in a row or two red doors in a row.

Entering through the blue door in the top left, can you find your way through the maze and exit through the red door in the top right?

Move exactly **2** matches to make these five squares into four squares of equal size, without any matches left over.

Move exactly 3 matches to make these three equilateral triangles into four equilateral triangles of any size, without any matches crossing or overlapping.

## Square in a Square

In the larger square, there is a smaller square formed by lines connecting each corner to the midpoint of one of the sides.

**Question**: What fraction of the larger square is the smaller filled square?

## Unclear Perimeter

What is the perimeter of the outer rectangle? All angles shown are right angles.

## Five’s a Crowd

Try to place five points in (or on the perimeter of) an equilateral triangle with side length 10, as to maximize the distance between the points.

What is the greatest possible distance between the two closest points?