- Define the circle's center at \((h, k)\) and a point on the circle at \((x, y)\).
- Construct a right triangle with vertices \((h, k)\), \((x, k)\), and \((x, y)\).
- Determine the horizontal leg as the distance between the x-coordinates: \(|x - h|\).
- Determine the vertical leg as the distance between the y-coordinates: \(|y - k|\).
- The lengths of the sides of the right triangle are \(\boxed{x - h, y - k}\).
Explanation
- Geometry of a circle and the right triangle construction
A circle with center \((h, k)\) and radius \(r\) is defined by the set of all points \((x, y)\) such that the distance between \((x, y)\) and \((h, k)\) is equal to \(r\). The distance formula provides the relation:
\(\sqrt{(x-h)^2 + (y-k)^2} = r\).
Squaring both sides gives the standard equation:
\((x-h)^2 + (y-k)^2 = r^2\).
To understand this geometrically, consider a point \((x, y)\) on the circle. We can construct a right-angled triangle by dropping a perpendicular from \((x, y)\) to the horizontal line passing through the center \((h, k)\). The vertices of this triangle are:
- The center of the circle, \(V_1 = (h, k)\).
- The projection of \((x, y)\) onto the horizontal line \(y=k\), which is \(V_2 = (x, k)\).
- The point on the circle itself, \(V_3 = (x, y)\).
- Calculating the leg lengths
The horizontal leg of this triangle connects \(V_1 = (h, k)\) to \(V_2 = (x, k)\). Its length is the distance between these points along the x-axis:
Horizontal leg length = \(|x - h|\).
The vertical leg of this triangle connects \(V_2 = (x, k)\) to \(V_3 = (x, y)\). Its length is the distance between these points along the y-axis:
Vertical leg length = \(|y - k|\).
These two segments are the legs of the right triangle, and the radius \(r\) is the hypotenuse, according to the Pythagorean theorem:
\((x - h)^2 + (y - k)^2 = r^2\).
Thus, the lengths of the sides (legs) are \(|x - h|\) and \(|y - k|\).
- Final Identification
Based on the derivation, the lengths of the legs of the right triangle are \(x - h\) (or \(|x - h|\)) and \(y - k\) (or \(|y - k|\)). Comparing this to the provided list:
- \(x - h\) is a correct choice.
- \(y - k\) is a correct choice.
The other options do not represent the lengths of these legs.
The final answer consists of the identified leg lengths.
\(\boxed{x - h, y - k}\)
Examples
Understanding the geometry of a circle is fundamental to navigation and engineering. For instance, if you are designing a rotating mechanical arm, the coordinates \((x, y)\) of the end of the arm at any time can be determined if you know the center \((h, k)\) and the length of the arm (the radius \(r\)). The right triangle formed by the legs \((x - h)\) and \((y - k)\) allows you to use trigonometry to calculate the exact position of the arm relative to its center, which is essential for precise control and movement.