Please help! I need to get this done quickly. :)

Question

Examine the diagram and information to answer the question. A circle in the coordinate plane has a radius of 5 and a center point (5,4). Point (x,y) lies on the circle. The triangle formed by points (5,4), (5,y) and (x,y) is a right triangle. 

What is the equation of the circle?

Step 1: Find the length of the vertical leg of the triangle.

Step 2: Find the length of the horizontal leg of the right triangle.

Step 3: Use the Pythagorean Theorem and ignore the absolute value since the expressions are being squared. 

Match the expression or equation to the steps used to find the equation of the circle.



The possible answers for the questions: 

1. (x-4)^2+(y-5)^2=5^2
2. (x-5)^2+(y-4)^2=5^2
3. |y-4|
4. |x-5|
5. |x-4|
6. |y-5|

Answer 1

We are asked to follow certain steps to derive the equation of a circle. So, to follow the argument you will need to first draw a picture. To do this you can follow the steps 1-5 below.

1. So the first step is draw a circle of radius 5 and centered on point (5,4).
2. Then choose any point (x, y) in the top-right quadrant of the circle. In reality, the point can be chosen anywhere but for easy communication without pictures let's follow this path.
3. Draw the radius from the center-point (5,4) to that point (x, y).
4. Mark the point (5, y). This will fall along the vertical diameter
5. Connect (5,4), (5,y) and (x,y) to complete the right triangle.

Now you can calculate the horizontal and vertical distances from the coordinates.

Horizontal distance between (x,y) and (5,y) = (x - 5)

Vertical distance between (5,y) and (5, 4) = (y - 4)

The hypotenuse is actually equal to the radius = 5

Following the pythagorean theorem you can now write the equation of a circle

(x-5)² + (y-4)² = 5²

This answer goes with choice #2 in your question.

Answer 2

The center-radius form of the circle equation is in the format (x – h)² + (y – k)² = r², with the center being at the point (h, k) and the radius being "r". 

Plugging the values given into this equation

(x – 5)² + (y – 4)² = 5²