5 km. 1. Open the sketch home.gsp provided by your teacher that shows the map above. * Construct a point P anywhere to represent James s home.

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1 uilding a Home Name James owns a small trout farm near the Drakensberg mountains, South frica, in the shape of a quadrilateral as shown. He wants to build his house so that it takes him the same time to travel to each of the four sides of his farm. Where should he choose to build his house P so that it is the same distance from all four sides of the farm? 11.1 km 6 km 10.1 km 60 deg 5 km Investigate efore you work in Sketchpad, draw a point in the drawing on the right that shows your best guess for the location of the reservoir. Label the point P. 1. Open the sketch home.gsp provided by your teacher that shows the map above. To measure the distance between a point and a line, select both point and line, and choose Distance in the Measure menu. * onstruct a point P anywhere to represent James s home. * Measure the distances from the point P to each of the four sides. * Drag point P and observe the four distance measurements. 2. Were you able to locate point P so that it is the same distance from all four sides? If so, how does the location you found in the Sketchpad sketch compare with your initial guess?

2 Simpler Problem How can you locate point P precisely without using trial and error and dragging? In problem solving it is often useful to look at a "simpler case" of a problem. In the original problem we wanted to find a point, which is equidistant (equal distances) from four sides. simpler case would be to look for a point (or points) equidistant from just two sides. D 11.1 cm P 6.0 cm 10.1 cm cm Scale 1cm:1km Distance P to D = 2.58 cm Distance P to D = 1.78 cm It might help to drag the two extra measurements off to the side or hide them, leaving behind only the measurements related to the two sides you have chosen to focus on. ontinue in the same sketch but focus only on two adjacent sides for now. * If necessary, drag point P so that it is equidistant from two adjacent sides. * Select point P and turn on Trace Point in the Display menu. * Drag point P slowly for a few centimeters, keeping it as equidistant as possible from the two adjacent vertices. 3. Describe as many properties as you can which relate the traced path to the angle formed by the two sides. hallenge: Use your observations from Question 3 to revisit the original problem with all four sides of the farm. ome up with an alternative way to search for a point equidistant from all four sides. Describe your construction method here. If you get stuck, continue onto the next page.

3 To construct an angle bisector in Sketchpad, first select three consecutive points defining the angle, ensuring that point at the vertex is in the middle. Then choose ngle isector in the onstruct menu. On the previous page, you should have found that there are infinitely many points that lie equidistant from two intersecting line segments (lines), and that they all lie on a straight line. Furthermore, from symmetry it should be clear that folding around Distance P to = 0.9 cm Distance P to = 0.9 cm Distance Q to = 1.8 cm Distance Q to = 1.8 cm Distance R to = 2.8 cm Distance R to = 2.8 cm Distance S to = 3.7 cm Distance S to = 3.7 cm Distance T to = 4.7 cm Distance T to = 4.7 cm this line of equidistant points, maps one side onto the other; therefore this line bisects the angle between the two sides. This line of equidistant points between two sides is called the angle bisector of the angle between two sides. * onstruct all four angle bisectors of the angles of the quadrilateral. 4. What do you notice about the four angle bisectors of this quadrilateral? * Drag point P to the point equidistant from all four sides. * onstruct a perpendicular from P to any one of the sides and construct its intersection with the side. P Q R S T * onstruct a circle with P as its center and the above intersection on the circumference. 5. Record what you observe about the other sides of the quadrilateral and explain why this must be true. More General Problem 6. Do you think you can always find a point equidistant from all four sides, no matter the shape or size of the quadrilateral? Explain. 7. onstruct a general dynamic quadrilateral with all four its angle bisectors. Drag any one of the vertices of the quadrilateral around the screen. What do you notice about the angle bisectors? 8. Do you still agree with your answer to Question 6? Explain.

4 Further Exploration 1. In a new sketch, construct a quadrilateral and a central point, so that the point is always equidistant from all four sides. Make the quadrilateral as general as possible. Make sure the central point is always equidistant from the sides no matter which points you drag. Explain your construction method. 2. The dynamic Sketchpad scale drawing of the farm is an example of a mathematical model that can be used to represent and analyze real-world situations. However, real-world contexts are complex and usually have to be simplified before mathematics can be meaningfully applied to them. What are some of the assumptions that have probably made in the problem at the start? 3. Suppose there is no equidistant point from the four sides of a quadrilateral (that is, the angle bisectors are not concurrent). Investigate what might be the best position to now build the home? an you mathematically explain why you think that would be the best position?

5 uilding nother House Name(s) Suppose James s farm was in the shape of a triangle as shown below. Where should he now choose his house P so that it is equal distances from all three sides? Investigate 1. efore you open up Sketchpad, draw a point on the map at the right that shows your best guess for the location of his house. Label the point P. To construct an angle bisector in Sketchpad, first select three consecutive points defining the angle, ensuring that point at the vertex is in the middle. Then choose ngle isector in the onstruct menu. * Open the sketch home2.gsp provided by your teacher that shows the map to the right. * onstruct the three angle bisectors of the sides of the triangle to locate the home correctly in your sketch. 2. How does the precise location compare to the location in your initial guess? More General Problem 3. Do you think you can always find a point equidistant from all three sides, irrespective of the shape or size of the triangle? Explain. 4. Drag any vertex of the triangle. What do you notice about the angle bisectors? Do you still agree with your answer in Question 3? Explain.

6 Explaining In the activity uilding a Home you found that the angle bisectors of a quadrilateral do not always meet in one point; in other words, the angle bisectors of the sides of a quadrilateral are not always concurrent. However, on the preceding page of this activity you should have discovered the rather surprising result that the angle bisectors of any triangle are always concurrent (at a point equidistant from all three sides). This point of concurrency is called the incenter of the triangle, since it is the center of the circle, which touches all three sides (the incircle). Why is the result always true for any triangle, but not for any quadrilateral? What is so special about the triangle? Let s explain why. If you came up with your own explanation for why the angle bisectors of any triangle are concurrent, compare it to that below. If not, work through the following. Let P be the point of intersection of two of your angle bisectors. We will show logically that this point P must also lie on the angle bisector of the third angle; that is, all three angle bisectors of a triangle always meet in the same point. P 5. Pick one of the two angle bisectors. What can you say about all the points on this bisector? 6. What can you say about all the points on the other angle bisector? 7. What can you therefore say about P, the point of intersection of both angle bisectors? 8. What can you therefore conclude about P and the angle bisector of the third angle?

7 Present your Explanation reate a summary of your explanation from Question 5-8. Your summary may be in paper form or electronic form, and may include a presentation sketch in Sketchpad. You may want to discuss the summary with your partner or group. Further Exploration 1. an the incenter of a triangle ever be outside or on the perimeter of a triangle? 2. onstruct a general quadrilateral D and any three of its angle bisectors. onstruct the intersection of two of these angle bisectors, and use it as a center to construct a circle, which always touches three of the sides. a. Drag the vertices of the quadrilateral until all three angle bisectors are concurrent. What do you notice? b. Drag the quadrilateral to a different shape until all three angle bisectors are again concurrent. lso construct the fourth angle bisector. What do you notice? c. In the space below, write a conjecture regarding your observations above. an you explain why it is true? an you generalize further to pentagons, hexagons, etc.? Discuss with your partner or group.