A Groups: Relationships between measures of arcs and central angles, inscribed angles. Hint: use a protractor to measure each central angle. Note: the measure of an arc is different from the length of an arc. Investigate the following diagram (1 on GSP) -Determine the lengths of KM and PN . -Draw in radii to points K, M, P, N. -Use a protractor to measure the central angles. K -Determine measures of arcs KM and PN. -Write a sentence to describe the arcs of congruent segments. B N M P Use point B as a center of dilation. Dilate the figure using several different scale factors. Repeat your measurements and decide whether or not your theory holds. Given that BC DE . Investigate the following diagram (2 on GSP) Determine the measures of the following: - BAC - BC - BDC - BEC - BFC B D E A F Write a sentence to describe a relationship between the measure of an inscribed angle and its intercepted arc. C Draw another point on the circle between E and F and label it G. Draw in chords from G to B and G to C. Determine the measure of BGC . Does your theory still hold? Is it possible for different inscribed angles to have the same measures? Investigate the following figure. (3 on GSP) H What is the measure of LHM ? How do you know? L D Q Draw in chords from D to L and D to M, From E to L and E to M, from F to L and From F to M, from G to L and from G to M. M E What are the measures of the inscribed angles LDM , LEM , LFM , LGM ? How do you know? F G Note: When you have a 90 degree inscribed angle, you also have a right triangle. The Pythagorean theorem may be useful in these situations! Things to share with your classmates: -Define central angle -Relationship of measure of central angle and measure of arc -Congruent segments have congruent arcs -Define inscribed angle -Relationship between inscribed angles and their intercepted arcs -Inscribed angles that are 90 degrees -Examples of different homework type problems B Groups: Relationships between measures of segments (chords, chord parts, secant segments, tangent segments) Describe why you think each of these segments/lines was named as they were. You will be investigating segment lengths formed by these segments and lines. In the following figure, describe lines CE and CG. (1 on GSP) Determine the lengths of CD, CE , CF , CG . E D C F G Triangles CDG and CFE are similar (CFE is reflected over CD , rotated, and dilated). The following figure shows the result after these transformations. Determine the scale factor using two different ratios: CD and CF ' CG . CE ' These scale factors should be the same! D F' C E' Rewrite the equation G CD CG = by first multiplying both sides by CF’. CF ' CE ' Multiply both sides of the equation by CE’. Write the result of your manipulation. This result is known as the secant segment rule. If two secant segments share the same endpoint outside a circle, then the product of the length of the secant and its external segment is the same as the product of the length of the other secant and its external segment. Write the secant segment relationship for the following diagram. Determine the value of the variable in each problem. C B A AC=20 BC=2 EC=10 DC=x D E Investigate the following figure. (2 on GSP) -Determine a list of transformations that would map triangle DEF onto triangle CBF. C B F A D -If triangles DEF and CBF are similar, the secant segment relationship applies here, as well. -Determine the lengths CF, BF, DF, and EF. E Use these lengths to check whether or not the secant segment relationship holds. Use the following figure to write out the chord segment relationship. If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. 14 7 4 x Investigate the following figure. (3 on GSP) A Determine the lengths AB, BC, CD. B C D Triangles ADC and DBC are similar. (reflection, rotation, and dilation). The following figure shows the result after these transformations. A D'' C B To determine the scale factor we could use CD " CB ' the ratios or . They should be the CA CD same. B' D CD " CB ' by first CA CD multiplying both sides by CA. Rewrite the equation Multiply both sides of the equation by CD. Write the result of your manipulation. This result is known as the secant tangent segment rule. If a secant and a tangent intersect at a point outside the circle, then the product of the length of the secant and its external segment is the same as the product of the length of the tangent segment and itself. Write the secant tangent segment rule for the given figure. Determine the value of the variable in each problem. x 5 4 Write a conjecture about two segments that are tangent to a circle from the same external point, as shown. Things to share with your classmates: -definitions of chord, secant, and tangent and where their names came from -secant segment rule and where it comes from -chord segment rule -secant and tangent segment rule -tangent segments from same external point -example homework problems C Groups: Relationships between measures of angles formed by secants, tangents, radii Describe why you think each of these segments/lines was named as they were. Using a compass and a straightedge (or GSP) draw 3 different sized circles and 1 tangent on each of the circle. Connect the center of each circle to its point of tangency on the circle. (A radius) Measure the angle formed between the radius and the tangent line segment. Write a sentence to describe these measurements. Do you believe these measurements will always hold? Why? Investigate the following figure. (1 on GSP) BC is tangent to circle A. C B Determine the measure of CBD . Determine the measure of BAD . A D The measure of BAD is also the measure of BD . Describe the relationship between CBD and BD . In general, if a tangent and chord intersect, the measure of the angle formed is equal to half the measure of the arc it intercepts. Determine the measure of CBE . Determine the measure of PSR Determine the measure of CBD . Investigate the following figure. (2 on GSP) Determine the measures of ABC and DBE . A mAC=134 What do you notice about these angle measures? B C Find the sum of the two arc measures. E D Do you notice any relationship between the angle measure and the sum of the arc measures? mED=110 Repeat the same investigation with the following figure. H 26 I J K 40 L The measure of an angle formed by two chords intersecting inside the circle is equal to the average of the two chords it intersects. Determine the value of x in the following problem. Investigate the following figure. (3 on GSP) F Determine the measure of AFB . C Find the sum of the two arc measures. 36 E A 98 B Find the difference of the two arc measures. Do you notice any relationship between the angle measure and the sum of the arc measures? The difference? Write a description of the relationship between an angle formed by two secants (or a secant and a tangent) outside the circle and the arcs it intercepts. Determine the value of x in the problem. Things to share with your classmates -measure of angle formed by a tangent and radius. Extension to using Pythagorean theorem. -relationship between angle formed with a tangent and chord and the arc it intercepts. -relationship between angle formed by chords and the arcs it intercepts. -relationship between angle formed by secants (or a secant and tangent) and the arcs it intercepts.