G12 Centers of Triangles

Transcription

1 Summer 2006 I2T2 Geometry Page Turn this page over and complete the activity with a different original shape. Scale actor Perimeter of Original shape Measuring Perimeter Perimeter of New shape How many times larger is the new perimeter? rea of Original shape Measuring rea rea of New shape How many times greater is the new area? What patterns do you notice? G12 enters of Triangles Objective: To construct the four centers of a triangle: the centroid (medians), circumcenter (perpendicular bisectors of the sides), incenter (angle bisectors), and orthocenter (altitudes).

2 Summer 2006 I2T2 Geometry Page entroid a. Open a new sketch. onstruct a triangle. b. Select two sides, under the onstruct menu, choose centroid Midpoints. onnect the midpoints to the opposite vertex to construct the medians. c. Locate the point of intersection. Label this point the centroid. Verify that the third median goes through this point. d. The centroid is called the center of mass of the triangle. If you were to cut out this triangle, you could balance it on the head of a pin placed at the centroid. 2. ircumcenter e. Using the same triangle as before, hide the three medians and the centroid point. (o not delete, we will need these later! The three midpoints should be left showing.) f. Through the midpoints, construct perpendiculars to two sides. circumcenter g. Locate the point of intersection. Label this point the circumcenter. Verify that the third perpendicular bisector goes through this point. h. To see the why this is called the circumcenter, select the circumcenter and point. Under the onstruct menu choose ircle by enter+point. You will see that the circumcenter is the center of the circle that circumscribes the triangle. s you can see, the circumcenter may be outside the triangle. 3. Incenter i. Using the same triangle as before, under the dit menu, choose Undo onstruct ircle. Hide the three perpendicular bisectors, the circumcenter point, and the three midpoints of the sides. (o not delete. Only triangle should be left.) j. Select an angle by clicking on the three points in order, then under the onstruct menu choose ngle incenter

3 Summer 2006 I2T2 Geometry Page 47 isector. Repeat this on a second angle. k. Locate the point of intersection. Label this point the incenter. Verify that the third angle bisector goes through this point. l. To see the why incenter J this is called the incenter, select the incenter and segment. Under the onstruct menu choose Perpendicular. Locate the point where the perpendicular from the incenter intersects segment. (The perpendicular line is the dotted line in the figure above. The point of intersection is labeled J in my construction.) m. Select the incenter point and the intersection point, then under the onstruct menu choose ircle by enter+point.

4 Summer 2006 I2T2 Geometry Page Orthocenter n. Using the same triangle as before, under the dit menu, choose Undo onstruct ircle, Undo onstruct orthocenter Intersection and Undo onstruct Perpendicular. Hide the three angle bisectors and the incenter point. (o not delete. Only triangle should be left.) o. Select vertex and, the side opposite. Under the onstruct menu, choose Perpendicular. Repeat for vertex and side. p. Locate the point of intersection. Label this point the orthocenter. Verify that the third altitude goes through this point. s you can see, the orthocenter may be outside the triangle. 5. The uler Segment This exploration uses the triangle and centers constructed above. Go under the isplay menu and choose Show ll Hidden. What you will see is a horrible mess. a. Starting with all the lines sticking out from the triangle, select about three at a time and hide them. ontinue until there are no lines or rays left sticking out from the triangle. b. Now select the three medians and the three midpoints of the sides. Hide them as well. c. You should now have only the four centers and the triangle remaining. To make them easier to see, relabel the orthocenter as O, the incenter as I, the circumcenter as, and the centroid as. d. rag a vertex of your triangle. Which centers appear to be collinear? re these centers collinear for any triangle? (Measure the sides to help determine the types of triangles.) re there any triangles where all four centers appear to be collinear? re there any triangles where all four centers are the same point? orthocenter incenter centroid circumcenter e. onnect the orthocenter O and the circumcenter of your triangle with a segment. This is called the uler Segment. Now drag around one vertex of the triangle. What do you notice? I O

5 Summer 2006 I2T2 Geometry Page 49 f. raw a horizontal segment across the top of your sketch. Select this segment and the top vertex of your triangle. Under the dit menu, choose Merge Point to Segment. g. Under the dit menu, choose ction utton and then slide to the right to nimation. dialog box should appear. Select medium as the speed. Then click on the OK button. The nimate Point button should appear on your sketch. L M O I nimate Point lick on the nimate button and observe what happens. lick on the button again to stop the animation. Geometer's Sketchpad comes with a file labeled Triangles.gsp that has interactive sketches involving the centers of the triangle and the uler line.

6 Summer 2006 I2T2 Geometry Page 50 ngle and arcs in a circle xploration #1 entral ngles Objective: To determine the relationship between the measure of a central angle and the degree measure of its intercepted arc. a. Using the circle tool, draw a circle in the work area. While the circle is selected, under the onstruct menu, choose Point on ircle. Label the center of the circle, the original point on the circle, and the second point on the circle as. b. hange back to the arrow tool. onstruct segments and. c. Measure. Measure arc by clicking on the circle, then points and. Under Measure choose rcngle to get the degree measure. d. Grab and move it around the circle. What do you notice about the relationship between the central angle and its intercepted arc? (The intercepted arc is the portion of the circle between the two sides of the angle.) e. Write a rule about these measures. xploration #2 Inscribed ngles Objective: To determine the relationship between the measure of an inscribed angle and the degree measure of its intercepted arc. a. Place a third point on the circle. The point should be labeled point. onstruct and. b. Measure. c. Grab and move it around the circle. Grab and move it around the circle. What do you notice about the relationship between the measure of the inscribed angle,, and the measure of its intercepted arc? d. Write a rule about these measures. m! = m on = m! = 23.15

7 Summer 2006 I2T2 Geometry Page 51 xploration #3 ngles formed by two chords intersecting in the interior of the circle. Objective: To determine the relationship between the measures of two vertical angles, formed by intersecting chords, and the degree measures of their intercepted arcs. a. Start a new sketch. Use the circle tool to put a circle in the work area. Label the center and the point on the circle. Place a point on the circle. b. Use the point tool to place a point inside the circle, not on. c. hange back to the arrow tool. onstruct lines and. d. onstruct the other two points where and intersect the circle. Label the intersection point for as and the intersection point for as. (See sketch below.) e. There are two pairs of vertical angles formed at. Measure one of the angles, then measure the two arcs m! = intercepted by the angle and by m on c its equal vertical angle. 1 = Remember to measure arcs you m on c 1 = must select the circle and the two endpoints of the arc. (Note: There are always two arcs of a circle with the same name, a minor arc (< 180 ) and a major arc (> 180 ). This program will always give the measure of the minor arc. If you want the major arc measure, subtract the minor arc measure from 360 or use a third point to name the arc.) f. Grab point and move it around. Grab point and move it around. What is the relationship between the measure of the angle formed by two chords intersecting inside the circle and the measures of the two intercepted arcs of the circle? (Hint: You might try adding the measures of the two arcs.) g. Write a rule about these measures.

8 Summer 2006 I2T2 Geometry Page 52 xploration #4 ngle formed by two secants intersecting outside the circle. Objective: To determine the relationship between the measure of an angle, formed by intersecting secants outside the circle, and the degree measures of the intercepted arcs. a. Start a new sketch. Use the circle tool to put a circle in the work area. Label the circle. Place a point on the circle. b. Use the point tool to place a point outside the circle. c. onstruct lines and. d. Label the other point where intersects the circle as and where intersects the circle as. (If one of the lines looks like a tangent, then move point or until you can see both points of intersection.) e. Measure and the two arcs it intercepts. On my sketch, I am measuring, arc and arc. m! = m on = m on = m on -m on = f. Grab point and move it around. Grab point and move it around. What is the relationship between the angle formed by two secants intersecting outside the circle and the two intercepted arcs of the circle? (Hint: You might try subtracting the smaller arc from the larger arc.) g. Write a rule about these measures. xploration #5 ngle formed by tangent and secant and angle formed by a tangent and chord. Objectives: To determine the relationship between the measure of an angle, formed by a tangent and a secant intersecting outside the circle, and the degree measures of the intercepted arcs. To determine the relationship between the measure an angle formed by the intersection of a tangent and a chord and the degree measure of the intercepted arc. a. Start a new sketch. Use the circle tool to put a circle in the work area. Label the circle. Place a point on the circle. b. onstruct a tangent to the circle at using the theorem that states a tangent is to the radius drawn to the point of tangency. o this by constructing radius. Select and point, then under onstruct choose Perpendicular Line.

9 Summer 2006 I2T2 Geometry Page 53 c. Place a point on the tangent line. Move point out away from the circle. d. onstruct line. Label the other point where line intersects the circle as. (If looks like a tangent, move point until it intersects the circle in two distinct points.) e. Measure and the two intercepted arcs. (Note: If one arc is > 180, you may have to place an additional point on that arc to name it. See sketch at right.) f. Grab point and move it around. Grab point and move it around. What is the relationship between the measure of the angle formed by a secant and a tangent intersecting outside the circle and the measures of the two intercepted arcs of the circle? (Hint: You might again try subtracting the smaller arc from the larger arc.) g. Write a rule about these measures. m! = m on = m on = m on -m on = h. onstruct. Measure and its intercepted arc. What do you notice? i. Write a rule about these measures. j. What kind of angle is? What is the relationship between m and the degree measure of arc? Verify this by measuring the angle and intercepted arc. m! = m on = 83.28

10 Summer 2006 I2T2 Geometry Page 54 xploration #6 ngle formed by two tangents Objective: To determine the relationship between the measure of an angle formed by two intersecting tangents and the degree measures of the intercepted arcs. a. Start a new sketch. Use the circle tool to put a circle in the work area. Label the circle. Place a point on the circle. b. onstruct tangents to the circle at both and using the theorem that states a tangent is to the radius drawn to the point of tangency. c. Move so that you can see the point where the two tangents intersect. m! = Label the point of intersection. Place an additional point on the circle on the major arc. d. Measure, the minor arc, and the major arc. e. Move around. What is the relationship between these measures? m on = m on = dditional explorations can determine the theorems involving the segment measures for intersecting chords, secants, and tangents.