Derive the standard formula of a circle. A radius [tex]$r$[/tex] is constructed as the hypotenuse of a right triangle as shown. What are the lengths of the sides of the right triangle that is formed? Check all that apply.
[ ] [tex]$h$[/tex]
[ ] [tex]$k$[/tex]
[ ] [tex]$h-k$[/tex]
[ ] [tex]$x-h$[/tex]
[ ] [tex]$x-y$[/tex]
[ ] [tex]$y-h$[/tex]
[ ] [tex]$y-k$[/tex]
Answer (1)
The center of the circle is ( h , k ) .
A point on the circle is ( x , y ) .
The lengths of the sides of the right triangle are x − h and y − k .
The equation of the circle is ( x − h ) 2 + ( y − k ) 2 = r 2 .
Explanation
Problem Analysis Let's analyze the problem. We are given a circle with radius r , and a right triangle is formed using this radius as the hypotenuse. We need to determine the lengths of the sides of this right triangle in terms of the coordinates of the circle's center and a point on the circle.
Determining Side Lengths Let the center of the circle be ( h , k ) and let ( x , y ) be any point on the circle. The horizontal side of the right triangle is the difference in the x-coordinates, which is x − h . The vertical side of the right triangle is the difference in the y-coordinates, which is y − k .
Conclusion Therefore, the lengths of the sides of the right triangle are x − h and y − k .
Examples
Understanding the equation of a circle is crucial in various fields, such as computer graphics, where circles are frequently drawn on screens. For instance, if you're developing a game and need to draw a target, you can use the circle equation to determine the position of each pixel on the circle's circumference, ensuring a smooth and accurate representation. This involves calculating the x and y coordinates relative to the center of the target, using the radius and the circle's equation.
