Unit 6 Lesson 1 Circle Geometry Properties Project Name____________________ Part A Look up and define the following vocabulary words. Use an illustration where appropriate. Some of this vocabulary can be found in the glossary of your textbook, others will need additional online resources. Vocabulary Word Definition Drawing Circle Tangent to a circle Secant to a Circle Chord Central angle Major arc Minor arc Inscribed angle Intercepted arc Sector Part B (Geogebra lab must precede this part) Using geogebra, complete each construction. Verify it meets the definition of the construction by showing angle measures or lengths (ie. If the line is a perpendicular bisector…show that the part being bisected has equal lengths on either side, AND show the angle is 90 degrees). With the measurements on the diagram, take a screen shot or snip your picture into this document under each bullet point. Construct the perpendicular bisector of a chord. (Start with a circle…draw in a chord…make the perpendicular bisector to that chord.) Place drawing here…. Construct a midpoint to a chord. Place drawing here…. Draw a line segment AB, and then construct a perpendicular line through a point on that line segment. Use the circle compass technique you tried early this year found on page 33 or follow this picture Place your drawing here…. Draw a line segment AB, Construct a perpendicular line through a point not on a the line AB. Use the circle compass technique you tried early this year found on page 34 or follow this picture Place your drawing here…. Part C Important Theorems Usually we start with an interesting drawing…show a pattern…and prove a result. This time we will go in reverse. Presented is a property or theorem that is the result of some proof based on construction. You will “deconstruct” the theorem. This means you will create a drawing to show this property is true. If indicated you will complete a drawing for different angles measures as well (acute, right, obtuse.) Take a screen shot or a snip the drawing and place it underneath the appropriate heading. 1. Tangent Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius at the point of tangency. (You will need to draw a circle, a tangent line, and show that the measure of the angle made between the circle and the radius at the point of tangency is 90 degrees) Place drawing here… a. Corollary: Page 408 #2, on angle bisectors. The corollary of the Tangent Theorem states that the line from the center of the circle to the intersection of the tangents, bisects the angle made by the tangents…in the diagram, OP, bisects angle APB. You will need to draw this and show angle APO is equal to angle BPO. Place drawing here… 2. Central Angle Theorem: The measure of a chord’s minor arc is equal to the measure of its central angle. (Use the following bullets to help you do this problem) Make a circle Put a point on the circle to create minor arc Connect the radii to the points on the circle Now use the sector tool to create the sector through the center and the two points. (click center of the circle, then B, then C Use the measure tool to find the central angle measure Use the measure tool to find the arc length of CB (when you click the arc length it will say arc length is. Measure a radius. Take the length of the arc and divide by its corresponding radius measure. You now have the arc measure in radians not an arc length.. Convert the arc measure to degrees by multiplying by 180 and dividing by 3.141592. The result should match the central angle measure. In my example drawing, the result should be 73.2 degrees. Repeat the drawing for all three types of angles ( acute, right, obtuse) Acute: Place drawing here… Right: Place drawing here… Obtuse: Place drawing here… 3. Center-Chord Theorem: A line through the center of a circle bisects a chord if and only if the line is perpendicular to the chord. (You must show this is true on two drawings, one for a chord that is a diameter as well as an arbitrary chord that is not a diameter.) Diameter: Place drawing here… Arbitrary chord: Place drawing here… 4. Inscribed Angle Theorem: The measure of an inscribed angle of a circle is half the measure of its intercepted arc. (must show for acute and right angle only) (***********Since, you showed the Central Angle Theorem to be true, you know the central angle has the same measure of the arc it intercepts. Therefore, to make showing this Inscribed Angle Theorem easier, you only need to show that the inscribed angle is half the central angle*************). Acute: Place drawing here… Right: Place drawing here… Obtuse: Place drawing here… 5. Inscribed Angle Corollary: Inscribed angles that intercept the same arc are congruent. (must show for acute and right angles only) Acute: Place drawing here… Right: Place drawing here… The next 3 conjectures are proof by the same construction…see page 402 question number 4 for insight into the drawing. You could copy and paste the same drawing for all three of these. 6. If a line contains the center of a circle and the midpoint of a chord, then the line is perpendicular to the chord. 7. The perpendicular bisector of a chord of a circle contains the center of the circle 8. If a line through the center of a circle is perpendicular to a chord, then the line contains the midpoint of the chord. Part D Application! 1. Complete problem 3 p. 399 2. Complete each of the attached circle worksheets Name__________________ Unit 6 Lesson 1 Circle Geometry Properties Practice For the last set of problems…refer to these two theorems on Secants and tangets. The examples here are extremely helpful . READ THESE FOR REAL, DO NOT JUST SKIM. .