proving two-tangent theorem using geogebra.docx
Download as DOCX, PDF0 likes76 views
This document presents the two-tangent theorem, which states that the two tangent segments to a circle from a point on the exterior are congruent and determine congruent angles with the segment from the exterior point to the center. The document provides background definitions, a general construction procedure using Geogebra in 9 steps, and a screenshot proving the theorem dynamically using Geogebra.
proving two-tangent theorem using geogebra.docx
- 1. Prepared by: KATHERINE E. BAUTISTA Theorem Background Construction o General Procedure Using Geogebra Proof of the Theorem o Dynamic Worksheet Using Geogebra Two- tangent theorem
- 2. I. THEOREM Two-Tangent Theorem The two tangent segments to a circle from a point on the exterior are congruent and determine congruent angles with the segment from the exterior point to the center. The theorem States that: AC = BC ∠CAO = ∠BAO II. BACKGROUND Definitions A tangent line to a circle is a line which is in the same plane as the circle and intersects the circle at one and only one point which is called the point of tangency. The line and the circle is tangent to this point. The exterior of the circle is the set of all points of the plane whose distance from the center is greater that the radius. Therefore, a point is on the exterior of the circle if the point is outside the circle. A tangent segment is a segment of a tangent line whose endpoints are the point of tangency and any other point on the tangent line. Some theorems related to tangent lines to circles Every line tangent to a circle is perpendicular to the radius drawn to the point of tangency.
- 3. The measure of an angle formed by two tangents of a circle intersecting at a point in the exterior of the circle is one-half the absolute value of the difference of the measures of the intercepted arcs. III. CONSTRUCTION. General Procedure Using GEOGEBRA Step 1. Start by creating a circle with a center O and a point D by clicking ‘Circle with center through point’ and renaming the points. Step 2. Make a point outside the circle to be the exterior point and name it A. Step 3. Make lines tangent to Circle O by clicking and choosing Step 4. Make the intersection of each tangent line and the circle. Name the points B and C, respectively. Steps 2 - 4
- 4. Step 5. Connect the points from A to C, A to B, O to C, O to B and A to O with segments. Step 6. Hide the lines by clicking the hide button beside the equations of the line in the algebra view pane. This will reveal the segments done in Step 5. Step 7. To prove the theorem, reveal the distances of segments AB and AC (the two tangent segments to a circle from a point on the exterior are congruent). It is also good to reveal the measures of the other segments formed. Step 8. Reveal the measures of the angles by clicking , then click the points determining ∠CAO and ∠BAO. This will prove that tangent segments AB and AC determine congruent angles with the segment from the exterior point to the center. It is also advisable to reveal the measures of ∠COA, ∠BOA, ∠ACO and ∠ABO to explore the theorem for corollaries. Steps 7 - 8
- 5. Step 9. Finally, insert texts and images to customize the works by. IV. PROOF OF THE THEOREM Dynamic Worksheet Using Geogebra To show the proof of the theorem, a screen picture of the solution using Geogebra is presented below: Two-Tangent Theorem Proved Using Geogebra
- 6. References: Alfarez, Merle S. and Alvia E. Lambino, MSA Geometry, Cainta Rizal, Philippines: MSA Publishing House.