The Inscribed Right Triangle Theorem can be used to justify why this construction works.
Consider OA, a radius of
Kriz is learning a graphic program. By default, the program shows segment AB and circle O. The segment's endpoint A can be moved anywhere outside of While endpoint B can be moved anywhere. An eye-like shape appears on the screen when AB is tangent to the circle. Give it a try!
Kriz can't quite place point B in position to see the eye-like shape appear. Help Kriz out!
How can a tangent line from a point outside of the given circle be constructed?
Since point A is a point outside B should be the point of tangency in order for AB to be tangent to the circle. On the example shape, by extending AB, it can be observed that B is the point of tangency.
Constructing a tangent from an outer point will help locate the point of tangency for a tangent drawn from A. Recall the steps in constructing a tangent.
In this case, point A is the outer point through which the tangent line is drawn. To get the example shape, move point A to the left as shown and then follow the steps.
As can be seen, the points where the circles intersect are the points of tangency. Therefore, point B should be on these points.
Suppose that a tangent line drawn from an outer point intersects a circle at A. When A is reflected across the line that passes through P and O, its image will also be a point of tangency for another tangent.
If AB and AC are tangent segments to then AB≅AC.
Consider two triangles.
These two triangles can be visualized in the diagram.
Because all radii of the same circle are congruent, it can be said that OB and OC are congruent. Moreover, △ABO and △ACO share the same hypotenuse OA. By the Reflexive Property of Congruence, OA is congruent to itself.
Combining all of this information, it can be said that the hypotenuse and one leg of △ABO are congruent to the hypotenuse and the corresponding leg of △ACO.
The points D, E, and F are the points where the segments touch the circle. If AB=12, BC=10, and CA=6, find AE.
From the graph, it can be seen that AD and AE are tangent segments with a common endpoint outside By the External Tangent Congruence Theorem, AE and AD are congruent.