Q1: Explain what is happening with the triangle edge and area measurements as you move each of the three vertices.

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1 GSP #3 Due: Thursday, July 13 FORMAT Write or type the questions embedded in the POW. Attach or embed sketches (only accepted if GSP sketches). Clearly label all pictures with GSP. Attach the provided GSP #3 cover sheet to the front of your work for turn in. Measurements and Calculations with GSP 1. Draw a long horizontal segment in the lower half of the screen. 2. Construct a point inches above the line. Select the point and the line and then use the Construct menu to construct a line through the point parallel to the line. 3. Construct a triangle between the lines. Use the lower line as the bottom edge of the triangle and construct a new point on the top line (not the one used to construct the line). Also construct two new points on the bottom segment. 4. Use the Construct menu to construct the three side segments for the triangle with these points. Label the vertices of the triangle. 5. Select the original line segment and the parallel line you created. Hide these using the Display Menu 6. You will now find the area of the triangle with the Calculate command. a. Assume the lowest edge is the base of the triangle. Select this edge and then choose the Length command from the Measure menu. b. Now select both the edge and the upper vertex and choose Distance from the Measure menu. This is the measurement of the length of the altitude. c. Deselect all the parts of the triangle and select both of the measurements. Choose Calculate from the Measure menu. The New Calculation pop-up calculator contains the measurements we selected as well as π and various functions. d. Click the Value drop down menu to select stored measurements. The asterisk in the keypad will indicate multiplication. Use this menu to compute the area of the triangle using the base and the height lengths. 7. You will now construct the interior of the triangle and measure the area of the triangle with the Measure menu. a. Select all three vertices of the triangle. Use the Construct menu to construct the Interior of the triangle and the Measure menu to measure the area of the triangle (interior). 8. Move each vertex of the sketch and watch the measurements change. Q1: Explain what is happening with the triangle edge and area measurements as you move each of the three vertices.

2 Altitudes of a Triangle with GSP An altitude of a triangle is a line segment extending from a vertex of the triangle to the opposite side, perpendicular to that side. In the following investigation you will discover some properties of altitudes. Part I Start a new GSP sketch 1. Construct an acute triangle ABC and its interior. 2. Construct a line perpendicular to segment AC through point B with the appropriate Construct menu command. 3. Construct D at the intersection of the new line and AC and then construct segment BD and hide the line. 4. Measure each angle of triangle ABC and the angle <BDC. 5. Label this sketch S1: This sketch has a real Altitude. PRINT S1 6. Drag B so that angle A or angle C becomes obtuse. Relabel this and call this S2: Altitude Adjustment. PRINT S2 RESPONSE including SKETCHES Q2: What happens to segment BD? Why do you think this happened? Part II 1. Create a new GSP sketch with a triangle whose sides are lines, not segments. 2. Construct all three altitudes using the same method as before except leave the altitudes as lines. 3. Label this sketch Triangle with a whole lot of Altitude. PRINT S3 RESPONSE including SKETCHES Q3: What do you notice about the altitudes? Are they parallel? Do the three different lines intersect in three different points? 4. The altitudes of the triangle are concurrent. That is, they intersect at a single point. This point is called the orthocenter of the triangle. RESPONSES including SKETCHES For the following questions, support each answer with a GSP sketch of a triangle, where the measures of the angles are shown. Q4: For what kind of triangle is the orthocenter interior to the triangle? Briefly explain your answer. (This is S4 Interior Ortho ) Q5: For what kind of triangle is the orthocenter exterior to the triangle? Briefly explain your answer. (This is S5 Exterior Ortho ) Q6: For what kind of triangle is the orthocenter on a vertex of the triangle? Briefly explain your answer. (This is S6 Vertex Ortho ) I

3 QUESTIONS: Q1: Explain what is happening with the triangle edge and area measurements as you move each of the three vertices. Q2: What happens to segment BD? Why do you think this happened? Q3: What do you notice about the altitudes? Are they parallel? Do the three different lines intersect in three different points? Q4: For what kind of triangle is the orthocenter interior to the triangle? Briefly explain your answer. (This is S4 Interior Ortho ) Q5: For what kind of triangle is the orthocenter exterior to the triangle? Briefly explain your answer. (This is S5 Exterior Ortho ) Q6: For what kind of triangle is the orthocenter on a vertex of the triangle? Briefly explain your answer. (This is S6 Vertex Ortho )

4 Name: Scoring Sheet GSP #3 QUESTION 1 QUESTION 2 QUESTION 3 QUESTION 4/SKETCH 4 QUESTION 5/SKETCH 5 QUESTION 5/SKETCH 6 SKETCH 1 : Clear, correct and : Lacking most SKETCH 2 : Clear, correct and : Lacking most SKETCH 3 : Clear, correct and : Lacking most

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