91Ó°ÊÓ

Understanding the radius of a square helps in solving various geometry problems. In this exercise, the radius is the distance from the center of the square to one of its vertices.

To visualize, imagine a square inside a circle where the center of the square and the circle coincide.

The radius is the hypotenuse of a right triangle formed by half the diagonal of the square.

• The sides of this right triangle are both the sides of the square.
• Given the side length of 10 inches, we use the Pythagorean theorem to find the hypotenuse.

The Pythagorean theorem states: \[ \text{hypotenuse}^2 = \text{side}^2 + \text{side}^2 \]
For our square:

• Hypotenuse = the radius we need
• side = 10 inches

Apply the theorem:
\[ \text{radius} = \frac{\text{side} \times \frac{\text{side}}{2}}{\text{side}} = \frac{10 \times 10/\root{2}}{10} = 10\frac{\root{2}} \]
This gives us a radius of approximately 7.07 inches.

The apothem of a square is the line segment from the center to the middle of one of its sides. It is a vital concept in geometry, especially in polygons.

Since a square is symmetrical, all sides and angles are equal. This symmetry simplifies the apothem calculation.

• Given a square with side length 10 inches
• The apothem is half the length of the side

Thus: \[ \text{Apothem} = \frac{\text{side}}{2} = \frac{10}{2} = 5 \text{ inches} \]
In any regular polygon, the apothem helps in area calculation and other geometric properties. For a square, it is a straightforward half-side.

The Pythagorean theorem is fundamental in geometry for solving problems involving right triangles. It states: \[ \text{a}^2 + \text{b}^2 = \text{c}^2 \] Where 'a' and 'b' are legs and 'c' is the hypotenuse.

In our exercise:

• The square's side lengths act as the right triangle's legs.
• Calculate the hypotenuse to find the radius.

For a 10-inch square, both legs 'a' and 'b' are 10 inches. The hypotenuse is the diagonal across the square.
Apply the formula:

• \[ 10^2 + 10^2 = \text{radius}^2 \]
• \[ 100 + 100 = \text{radius}^2 \]
• \[ 200 = \text{radius}^2 \]
• \[ \text{radius} = \root{200} = 10\root{2} \]

Therefore, the Pythagorean theorem confirms the radius is approximately 14.14 inches for a side of 10 inches.