Question

Please help, i know you have to use pythagoras and the hypotenuse is 8.44 but i dont know how to get one of the sides

Answer 1

Answer: 5.97 cm

======================================================

Step-by-step explanation:

Check out the diagram below.

I've defined the following points:

• A = center of the circle, and center of the square
• B, C, D, and E = vertices of the square that are on the circle
• F = midpoint of segment CD

Now focus on triangle ACF. This is an isosceles right triangle with legs of 0.5x each. The hypotenuse is the radius of the circle. Let R be that unknown radius.

The area of the circle is given to us. It is 56 square cm. Use this to find the value of R.

`A = \pi*(\text{radius})^2\\\\56 = \pi*R^2\\\\R^2 = 56/\pi\\\\R = √(56/\pi)\\\\R = 4.22200825\\\\`

This value is approximate. I used the calculator's stored version of pi to get the most accuracy possible.

------------

We now know the hypotenuse of triangle ACF.

Use Pythagoras (aka Pythagorean Theorem) to have the following steps:

`a^2+b^2 = c^2\\\\(0.5\text{x})^2+(0.5\text{x})^2 = 4.22200825^2\\\\0.25\text{x}^2+0.25\text{x}^2 = 17.82535366\\\\0.5\text{x}^2 = 17.82535366\\\\\text{x}^2 = 17.82535366/0.5\\\\\text{x}^2 = 35.65070732\\\\\text{x} = √(35.65070732)\\\\\text{x} = 5.97082132708726\\\\\text{x} = 5.97\\\\`

The final answer is 5.97

With the exception of 0.5 and 0.25, each decimal value mentioned is approximate.

Keep in mind that rounding to 3 sig figs means that we're rounding to 2 decimal places. The units digit is 1 sig fig, and the 2 decimal places give a total of 1+2 = 3 sig figs.

In short, we're rounding to the nearest hundredth.

--------------------

Check:

x = 5.97 is the square's side length.

It divides in half to get x/2 = 5.97/2 = 2.985 which gives each leg of triangle ACF.

Use a = 2.985 and b = 2.985 in the Pythagorean Theorem to find that c = 4.22142748 approximately. This is the approximate radius of the circle.

Use that radius to find the area of the circle.

A = pi*r^2

A = pi*(4.22142748)^2

A = 55.984594705958

A = 56.0

I'm rounding the area to 3 sig figs to keep consistent with the rounding done to x. We arrive at the circle area of 56 square cm, which helps confirm the answer is correct.