A circle is the set of all points that are equidistant from a given point. The given point is usually called the center of the circle. The distance between the center and any point on the circle is called a radius.
In a circle, segments can be drawn from the circle and the center. Depending how the segments are drawn, they have different names and properties.
Notice that a radius is half of the diameter and not a chord. Additionally, there are two noteworthy lines that intersect a circle.
Name the lines and segments in the circle where is the circle's midpoint.
To begin, notice that and are the only lines in the figure. Since the others are segments, it is possible that they are chords. Recall that a chord is a segment between two points on the circle. Thus, we can identify three chords in this circle.
Therefore, the segments and are all chords. Since passes through the circle's center, it's called a diameter. The segment is a part of the line A line that intersects a circle at two points is called a secant. Thus, is a secant.
The segment has endpoints at the circle and its center. Such segment is called a radius and is half the length of a diameter. The last line, touches the circle at the point and is perpendicular to the radius
Therefore, is a tangent to the circle. We have now identified all lines and segments intersecting the circle.
|Tangent to Circle Theorem|
According to the Perpendicular Postulate, there is only one segment from that is perpendicular to It will be shown, using contradiction, that must be this segment.
Suppose that is not perpendicular to It follows then that there exists another segment from perpendicular to Call this segment
Because a right angle is created at their intersection. As a result, which measures units, is the hypotenuse — and the longest side — in the right triangle Therefore, the length of is shorter than Notice that contains a radius of circle Because part of the segment lies outside the circle, the unknown portion can be assigned another variable,
By the Segment Addition Postulate, The relationship between and can be written as an inequality.
By subtracting on both sides, the inequality states that Since a length cannot be less than the assumption that is not perpendicular to must be false. Therfore, is perpendicular to the tangent
This can be summarized in the following two-column proof.
|Draw line segment||Construction of triangle|
|Express triangle's hypotenuse|
|Express leg of triangle|
|Hypotenuse longer than leg|
|Length must be positive|
|Proof by contradiction|
|Similar Circles Theorem|
Two figures are similar if there exists a similarity transformation that maps one onto the other. Consider circles and as shown.
Circle can be mapped onto circle through a composite transformation. First, it can be translated left and down so that the midpoints overlap. Then the size of circle can be enlarged through a dilation.
To determine the scale factor, of the dilation, divide the radius of with the radius of
Tha radius of is then multiplied with the scale factor, which maps to Thus, because a similarity transformation exists that maps circle onto circle Therefore, all circles are similar.
This can be summarized in the following flowchart proof.