Question

A circle with the equation (x + 3)^2 + (y - 2)^2 = 25 is reflected over the line x = 2. What is the equation of the image? (x - 3)^2 + (y + 2)2 = 25 (x - 7)^2 + (y + 2)^2 = 25 (x - 7)^2 + (y - 2)^2 = 25 (x - 3)^2 + (y - 2)^2 = 25 A square is inscribed in a circle that has a radius of 2√2 inches. What is the length of the side of the square? 8√2 in. 8 in. 4√2 in. 4 in.

Answer 1

Part A
It's important to first understand how to form the equation of a circle.
If our center is (h, k) and our radius is r, the circle's eqn. is (x-h)² + (y-k)² = r².

Let's interpret our circle here.
It has the eqn. of (x+3)² + (y-2)² = 25.
The center must be (-3, 2) and the radius must be 5.
(to get that plus inside the (x-h) h must be negative for it to cancel)

When this circle gets reflected, the radius won't change, but the center will.
If we reflect our center (-3, 2) over x = 2, it would end up at (7, 2).
Let's show this in our new equation.

(x-7)² + (y-2)² = 25

Part B
Draw yourself a diagram for this question.
Draw a circle, a square inscribed within it, and the square's diagonals.
Note how the diagonals of the square are diameters of the circle.
If the radius of the circle is 2√2 inches, the diameter must be 4√2.

Now just consider the square. Its diagonal is 4√2 inches long.
Of course, this diagonal along with two sides of the square creates a 45-45-90 triangle, which has opposite sides in proportion of x, x, and x√2, respectively.
The 4√2 being opposite the right (90) angle, we can consider it x√2 in this proportion. This means that x, which would then be the side of the square, is 4.