Question

Many younger children like to play a game similar to baseball called tee-ball. Instead of trying to hit a ball thrown by a pitcher, the batter hits the ball off a tee. To accommodate younger children, the bases are only 40 feet apart. Find the distance between home plate and second base.

Short answer

The distance is approximately 56.57 feet.

Step-by-step solution

Identify the Shape of the Field

In tee-ball, the bases form a square since the field mimics a baseball diamond layout. Each side of this square is the distance between two successive bases, which is 40 feet.

Determine the Diagonal of the Square

The diagonal of this square is the distance from home plate to second base. Since home plate, first base, and second base form a right triangle, we can use the Pythagorean theorem to find this diagonal (hypotenuse).

Apply the Pythagorean Theorem

The Pythagorean theorem states that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two other sides. Here, both sides of the right triangle are 40 feet, so: \[ a^2 + b^2 = c^2 \] where \( a = 40 \) feet, \( b = 40 \) feet, and \( c \) is the diagonal (hypotenuse).

Calculate the Diagonal Length

Substitute the known values into the equation from the Pythagorean theorem:\[ (40)^2 + (40)^2 = c^2 \]\[ 1600 + 1600 = c^2 \]\[ 3200 = c^2 \]Take the square root of 3200 to find \( c \):\[ c = \sqrt{3200} \]\[ c \approx 56.57 \] feet.

Conclusion

Therefore, the distance between home plate and second base is approximately 56.57 feet.

Key concepts

Understanding Geometry in Tee-Ball

In the context of tee-ball, the layout of the playing field is an excellent introduction to understanding basic geometry concepts.

• The field is set up in the shape of a square.
• Each side of the square represents the distance between two consecutive bases, which we know to be 40 feet.

This setup helps form a two-dimensional shape where all sides are equal in length. The bases form a clear pathway that defines the playing area and helps in understanding shapes and measurements. Recognizing shapes, like squares, is crucial in geometry because it opens the pathway to more complex geometric reasoning and problem-solving.
Understanding how to visualize this field as a square will help connect the distances between bases and apply geometric concepts such as finding distances or perimeters.

Right Triangle and Its Role

In tee-ball, while the bases form a square, the diagonal measurement within the square forms a right triangle. A right triangle is a triangle where one angle measures 90 degrees.

• This naturally occurs when two sides meet perpendicularly, as they do in the field setup.
• The right triangle formed from home plate to first base to second base helps apply significant mathematical principles, like the Pythagorean theorem.

The knowledge of right triangles is fundamental in geometry because it leads into understanding relationships between side lengths and angles. Specifically, in a right triangle, the longest side, called the hypotenuse, can be determined using these relationships. This is a crucial understanding for calculating distances efficiently.

Square and Its Diagonal

A square is a special type of rectangle where all four sides are equal, and like any rectangle, it has diagonals that bisect each other at right angles.

• In our tee-ball diamond, the square comprises four equal sides, each measuring 40 feet.
• We aim to calculate the diagonal, which in this setup, extends from home plate to second base.

The diagonal of a square doesn't just split it into two right triangles; it also becomes a key factor in distance measurements. The diagonal can be found using the Pythagorean theorem, indicating its significance in geometry.
The square acts as a fundamental building block, offering insight into numerous other shapes and equations in geometry. Through the understanding of squares and their properties, students learn to tackle a wide range of mathematical problems.