The External Tangent Congruence Theorem states that if two tangent segments from a common external point are congruent, then the segments will be equal or congruent. For example, if SR and ST are tangent segments, then SR = ST.
To prove the External Tangent Congruence Theorem, we can use the tangent segment theorem formula which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. For example, in the case of SR and ST, if we let SR = x and ST = x, with R = 5 and T = 11, we can use the formula a² + b² = c² to solve for x. In this case, 5² + 11² = 13², which simplifies to 25 + 121 = 169, and thus x = √(169) = 13.
An example of the External Tangent Congruence Theorem in action is when we have the tangent segments from a point outside a circle being congruent, and we can use the formula to prove their congruence.
The Tangent Line to Circle Theorem states that if a line is tangent to a circle at a point, then the line is perpendicular to the radius at that point.
To prove the Tangent Line to Circle Theorem, we can use the formula to find the radius of the circle using the Pythagorean Theorem. We can set up the equation r² + 80² = (r+50)² and use the formula to solve for r in the case where the line is tangent to the circle.
An example of the Tangent Line to Circle Theorem is when we have a line tangent to a circle at a given point, and we can use the formula to determine the radius of the circle.
The Congruent Central Angles Theorem states that in the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.
For example, in the case of two circles being congruent, their central angles will determine the congruence of their arcs. However, if the arcs do not have the same measure, then they are not congruent as per the theorem.
The Congruent Circles Theorem states that two circles are congruent if and only if they have the same radius. An example of this is when we have two circles with the same radius, which makes them congruent according to the theorem.
The Equidistant Chords Theorem states that in the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. An example of this is when two chords are equidistant from the center of a circle, making them congruent according to the theorem.
Overall, these theorems and formulas are essential tools in solving problems related to tangent segments, tangent lines to circles, and congruent central angles in geometry. They provide a systematic way to prove and apply the relationships between segments, lines, and angles in circles.
