Materials Voyage 200/TI-92+ calculator with Cabri Geometry Student activity sheet Shortest Distance Problems. Introduction

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1 s (Adapted from T 3 Geometry/C. Vonder Embse) Concepts Triangle inequality Distance between two points Distance between a point and a line Distance between two points and a line Perpendicularity Materials Voyage 200/TI-92+ calculator with Cabri Geometry Student activity sheet Shortest Distance Problems Introduction What is meant by the shortest distance? In this activity you will investigate three cases of shortest distance : a. the shortest distance between two points, A and B b. the shortest distance between a line, n, and a point, A, not on line n c. the shortest total distance from a point, A, to a line, n, to another point, B, on the same side of line n. You will be encouraged to verbalize the conditions that must be true for three segments to form a triangle and then to organize your thinking into a general rule known as the Triangle Inequality Theorem. In this activity you will use the Cabri Geometry tools to draw a point, segment, and line (F2: Points and Straight Objects), construct a reflection of a given object (F5: Transformations), measure distance and length, calculate other measures, and check the property of objects (F6: Measurement), and to label objects (F7: Display). PTE: Dynamic Geometry Page 1

2 Student Activity Sheet Case 1: The shortest distance between two points A. Construct an example to illustrate this. 1. Draw points A and B using the Point tool (F2 menu). 2. Construct AB using the Segment tool (F2 menu). 3. Measure the length of AB using the Distance and Length tool (F6 menu). 4. Draw point C not on AB and construct and measure AC and BC. 5. Use Calculate (F6 menu) to compute the sum of AC + BC (Figure 1). 6. Drag point C around the screen and compare the length of AB to the sum AC + BC (F6 menu). Figure 1 B. Note your observations. If C is not on AB or an extension of AB, then the three points form ABC. 1. What is the shortest distance between two points? 2. Under what conditions will three segments form a triangle? 3. How can you use this information to explain that AB must be shorter than the sum AC + BC? PTE: Dynamic Geometry Page 2

3 Case 2: The shortest distance between a line and a point not on the line A. Construct an example to illustrate this. 1. Draw a line and a point on the line using the Line tool (F2 menu). 2. With the Label tool (F7 menu), label the line n and the point. 3. Using the Segment tool (F2 menu) draw AB such that point A is not on line n (Figure 2). 4. Measure the length of AB using the Distance and Length tool (F6 menu). You may wish to increase the number of decimal places in your measurement to detect small difference (change the Display Figure 2 Precision in the Format tool in the F8 menu). 5. Drag point B along line n until the length of AB is as short as possible. B. Use your observations to answer the following questions. 1. What is the shortest distance between a line and a point not on the line? 2. Make a conjecture that seems to always be true about this construction. Using the Check Property tool (F6 menu), test your conjecture. 3. How else might you have tested to check for perpendicularity? Do you think that you have proved your conjecture? PTE: Dynamic Geometry Page 3

4 Case 3: The shortest total distance from a point to a line to another point on the same side of the line A. Construct an example to illustrate this. 1. Draw line n and a point C on line n (F2 menu). 2. Draw points A and B on the same side of line n but not on line n (F2 menu). 2. Draw and measure AC and BC (F6 menu). 3. Measure AC and BC. Use the Calculate tool (F6 menu) to sum the total distance from A to C to B (Figure 3). 4. Drag point C along line n until the sum AC + BC is as small as possible. 5. Drag point A, point B or both and then move point C to minimize the sum AC + BC. Figure 3 B. Use your observations to answer the following questions. 1. What is the shortest total distance from a point to line to another point on the same side of the line? 2. Explain any patterns you see. [Hint: reflect point A across line n to point A' and compare measurements.] Further Explorations: 1. Use the results of Case 3 of this activity to describe the path that a billiard ball follows as it rebounds off of a cushion. 2. What is the shortest distance between three points? PTE: Dynamic Geometry Page 4

5 Teacher Notes Introduction This exploration should be done as a classroom demonstration to establish the Triangle Inequality Theorem. Encourage students to verbalize the conditions that must be true for three segments to form a triangle then have them organize their thinking into a general rule. Instructions Case 1 The basic conjecture of this activity is that for a figure to be a triangle, the sum of the length of two of the sides must be larger than the length of the third side. The proof of this conjecture leads to the Triangle Inequality Theorem which implies that the segment AB must be shorter than the sum AC + BC for any position of point C not on AB or an extension of AB (Figure 4). If point C is on AB, then no triangle is formed since three collinear points do not form a triangle. In this case, AB = AC + BC (Figure 5). Figure 4 Figure 5 Case 2 The basic conjecture of this activity is that the shortest distance from a point to a line is a segment perpendicular from the point to the line. Two methods are given to show this concept. It is advisable to change the Display Precision (9:Format under F8 menu) to expand the number of decimal places in the numeric value for AB so fine differences can be seen. One method to show this concept is to measure the angle(s) formed by line n and segment AB. When point B is dragged so that AB is perpendicular to n, the angle(s) will measure 90. Another method is to use Perpendicular from the Check Properties menu under F6 to test if line n is perpendicular to AB. Have the students drag point B along line n and observe how the length of AB changes and becomes smaller as AB approaches the position perpendicular to line n. When the line and the segment are perpendicular, the message "Not perpendicular" changes to "Perpendicular" and the length of AB is minimized (Figures 6 and 7). PTE: Dynamic Geometry Page 5

6 Figure 6 Figure 7 Case 3 The basic conjecture of this activity is that the shortest total distance from a point to a line to second point on the same side of the line is equal to the distance from the first point to the reflection of the second point across the line. Increase the number of decimal places to 5 or 6 (9:Format in the F8 menu) in the sum AB + BC so that small changes can be seen as point C is dragged along line n. When the distance AB + BC is minimized, the angle between the segments and the line will be approximately equal (Figure 8). Indicate to the students that exact equality is not always possible because of the discrete nature of the computer or calculator screen. This line of exploration should suggest to the students that the two angles are probably equal. Figure 8 Figure 9 Using the Reflection tool (F5 menu), reflect point A across line n to point A' and draw the segment A' C (Figure 9). The newly formed angle between n and the new segment A' C is exactly equal to the angle between the line n and segment AC. Measure A' C and drag point C along line n to establish that AC = A' C for any position of point C (Figure 8). Since AC = A' C, then A' C + BC = AC + BC. To minimize the sum AC + BC, the sum A'C + BC must be minimized. Since the shortest distance between two points is a straight line, the segment A' B must be the shortest distance between points A' and B. But, by the Triangle Inequality Theorem, A' C + BC > A' B, unless point C lies on A' B (Figures 10 and 11) When point C is on A' B, A' C + BC = A' B. The point of intersection of segment A' B and line n is the shortest total distance from point A to line n to point B. The same argument holds if point B is reflected across line n. PTE: Dynamic Geometry Page 6

7 Figure 10 Figure 11 PTE: Dynamic Geometry Page 7

TImath.com. Geometry. Triangle Inequalities

TImath.com. Geometry. Triangle Inequalities Triangle Inequalities ID: 9425 TImath.com Time required 30 minutes Topic: Right Triangles & Trigonometric Ratios Derive the Triangle Inequality as a corollary of the Pythagorean Theorem and apply it. Derive