Question

1 Linda is designing a circular piece of stained glass with a diameter of 7 inches. She is going to sketch a square inside the circular region. To the nearest tenth of an inch, the largest possible length of a side of the square is

Answer 1

The largest side length of a square that fits inside a circle with a 7 inch diameter is approximately 4.95 inches, calculated using the Pythagorean theorem.

Step-by-step explanation:

Linda is creating a circular stained glass piece and wants to know the largest possible side length of a square that can fit inside it. The circle has a diameter of 7 inches, which means the radius is 3.5 inches. The largest square that can fit inside the circle is one where the diagonal of the square is equal to the diameter of the circle.

Using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):

a2 + b2 = c2. Since the square's diagonal is the hypotenuse and the sides of the squares are equal (a = b), the equation can be simplified to:

2a2 = c2

Filling in the diameter of the circle for c (7 inches), and solving for a gives us:

2a2 = 72

a2 = 49 / 2

a = \(√(49 / 2)\)

a ≈ 4.95 inches

Therefore, the largest side length of the square that can fit inside a circle with a diameter of 7 inches is about 4.95 inches.

Answer 2

To sketch a square inside a circular region such that the length of the square is of the largest possible value, we note that the four vertices of the square will touch the circumference of the circle and thus the length of the diagonal of the square is equal to the diameter of the circle.

Thus, the length of the diagonal of the square = 7 inches.

Let the length of the sides of the square be s, then using pythagoras theorem

`s^(2) + s^(2) =7^2 \\ 2s^2=49 \\ s^2=24.5 \\ s= √(24.5) \approx 4.95`

Therefore, the largest possible length of a side of the square to the nearest tenth of an inch is 4.95 inches.