GSP 3: Triangles 1. Triangle Centers There are many types of triangle centers. Four common triangle centers are: Incenter; Circumcenter; Centroid; Orthocenter. • The Incenter is the point located at the intersection of angle bisectors. • The Circumcenter is the point located at the intersection of perpendicular bisectors of the sides. • The Centroid is the point located at the intersection of the medians. • The Orthocenter is the point located at the intersection of the altitudes. (Name tab: 1a-­‐d) a. Incenter: Construct an arbitrary and flexible triangle ABC. Construct all three angle bisectors (change their Display to easily distinguish them from the sides). Label the intersection point I. Label this construction “Incenter.” Do you need to construct all three angle bisectors to determine the Incenter? How many are needed? Why? b. Circumcenter: Construct an arbitrary and flexible triangle ABC. Construct all three perpendicular bisectors of the sides (change their Display to easily distinguish them from the sides). Label the intersection point M. Label this construction “Circumcenter.” Do you need to construct all three perpendicular bisectors of the sides to determine the Circumcenter? How many are needed? Why? c. Centroid: Construct an arbitrary and flexible triangle ABC. Construct all three medians (change their Display to easily distinguish them from the sides). Label the intersection point D. Label this construction “Centroid.” Do you need to construct all three medians to determine the Centriod? How many are needed? Why? d. Orthocenter: Construct an arbitrary and flexible triangle ABC – use lines, not line segments, and be sure to make the vertices the points used to construct each line. Construct all three altitudes (change their Display to easily distinguish them from the sides). Hide the lines and then overlay segments for the triangle sides. Label the intersection point O. Label this construction “Orthocenter.” Do you need to construct all three altitudes to determine the Orthocenter? How many are needed? Why? (New tab: 1e) e. Construct a flexible equilateral triangle. Verify your ∆ is equilateral with GSP measurements. Construct the Incenter, Circumcenter, Centriod and Orthocenter. Move around your triangle. Does it appear that any of the centers’ locations are related to each other? Describe. (New tab: 2) 2. Euler Line. a. Construct an arbitrary and flexible triangle ABC. Construct the Circumcenter, label it CC, and hide its construction lines. Construct the Centroid, label it CD, and hide its construction lines. Construct the Orthocenter, label it OC, and hide its construction lines. Move around your triangle. Does it appear that any of the centers’ locations are related to each other? Describe. b. Construct one line that passes through all three of the centers. This line is called the Euler Line! c. Copy your construction from part a 7 times (below the original). d. Label each of the seven copies a different triangle type. e. For each ∆, move the vertices around so that it matches its description (by visual inspection – no GSP measures required) f. Can the Euler line still be formed for each type of triangle? If there is a triangle where it cannot be formed then say which and describe why. 3. Triangle Congruencies We have a great, informal, way of determining when two polygons are congruent, that is, when we overlay them and they match up exactly. A slightly more formal way to think about polygon congruency is that two polygons are congruent when their corresponding sides are congruent and their corresponding angles are congruent. For example in the picture below ∆ABC ≅ ∆DEF and we can see that •
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Corresponding sides are congruent: AB ≅ DE and BC ≅ EF and CA ≅ FD Corresponding angles are congruent: ∠A ≅ ∠D and ∠B ≅ ∠E and ∠C ≅ ∠F C
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A unique property of triangles is that we do not have to test all six congruencies in order to know that two triangles are congruent (or not). F
In the following activities you will investigate that various triangle congruency tests. In these activities you will be starting with a ∆ and building a ≅ ∆, that thinking is the same as when you have two ∆s and you are testing if they are ≅ or not. (New tab: 3a) a. Side-­‐Side-­‐Side ∆ ≅ test (SSS) SSS Construction: “Copy” each side. • Congruent segments have the same length. When constructing ≅ segments with physical tools we use a compass and straight edge (not a ruler – that would only give us an approximation). In GSP we will use the Circle Tool to construct congruent, corresponding segments (we will not do anything with the angles). 1. Construct arbitrary ∆ABC. 2. Construct a point off of AB and name it A’. Select A’ and AB (but not its endpoints). Construct a circle centered at A’ with radius ≅ to AB by using the Construct menu -­‐> Circle by Center+Radius. 3. Attach a point onto the circle and name it B’. Then construct A'B' . Use the display menu to make the circle “Thin” and keep the line segments “Medium.” 4. Move around the points, A, A’, B, B’ and observe that AB ≅ A'B' no matter how you change things. 5. Construct a circle at A’ with center congruent to AC (make the circle “Thin”). Vertex C’ will be on that circle somewhere, but we do not know where yet – so DO NOT put a point on that circle yet. 6. Construct a circle at B’ with center congruent to BC (make the circle “Thin”). Vertex C’ will be on that circle somewhere. There are two points where we could place C’. Describe where to put C’ and why. Then construct and label C’ (Hint: Using the Selector to click on intersections creates the exact point of intersection.) 7. Construct A'C' and B'C' . Move ∆A’B’C’ around so that it overlays on ∆ABC and matches up exactly. Hopefully they match up and you have informally proved that they are congruent. 8. When given two ∆s, describe how to use the SSS test to determine if they are ≅ or not. (New tab: 3b) b. Side-­‐Angle-­‐Side ∆ ≅ test (SAS) SAS Construction: “Copy” two sides and the angle between them. • Congruent angles have the same measure. When constructing ≅ ∠s with physical tools we use a compass and straight edge (not a protractor – that would only give us an approximation). In GSP we will use the Circle Tool to construct congruent, corresponding segments (we will not do anything with the angles). 1. Construct arbitrary ∆ABC (keep the line segments “Medium”). 2. Construct a point off of AB and name it A’. Construct a circle centered at A’ with radius ≅ to AB . 3. Attach a point onto the circle and name it B’. Then construct A'B' . Keep the circle “Thin” and the line segments “Medium.” 4. Next, you will construct an angle on A'B' at A’ that is ≅ to ∠A. If you were using physical tools, you could only use a compass and straight edge, however in GSP you can use a Transformation to do this: a. Mark ∠BAC for transformation by selecting B, A, C, in order. Then use the Transform menu -­‐> Mark Angle. b. Double click on A’ (you should see it mark for transformation). c. Select only B’. d. Use the Transform menu -­‐> Rotate -­‐>Marked Angle (click Rotate). e. Construct a Ray starting at A’ through the rotated point you just made. f. Change the size of ∠A and you should see ∠A’ change correspondingly. 5. Construct a circle at A’ with center congruent to AC (make the circle “Thin”). Vertex C’ will be on that circle somewhere, in only one possible location. Describe where to put C’ and why. Then construct and label C’ (Hint: Using the Selector to click on intersections creates the exact point of intersection.) 6. Construct A'C' and B'C' . Move ∆A’B’C’ around so that it overlays on ∆ABC and matches up exactly. Hopefully they match up and you have informally proved that they are congruent. 7. When given two ∆s, describe how to use the SAS test to determine if they are ≅ or not. (New tab: 3c) c. Angle-­‐Side-­‐Angle ∆ ≅ test (ASA) ASA Construction: “Copy” two angles and the side between them. 1. Construct arbitrary ∆ABC (keep the line segments “Medium”). Switch 3 and 4
2. Construct a Ray off of ∆ABC and name its endpoint A’ and label the other construction point D. (Make the ray “Thin”). !!!!"
3. Construct an angle on ray A'B' at A’ that is ≅ to ∠A (refer to #4 in the previous construction). [Note: Move point D out of the way – you will only use D at the end to help move your construction around.] AB . Construct the intersection 4. Construct a circle centered !!!!" at A’ with radius ≅ to of that circle and ray A'B' and name it B’ (make the circle “Thin”). Overlay a “Medium” segment, A'B' . !!!!"
5. Construct an angle on ray A'B' at B’ that is ≅ to ∠B (refer to #4 in the previous construction). (Keep the rays “Thin.”) 6. Vertex C’ can only be in one possible location. Describe where to put C’ and why. Then construct and label C’ (Hint: Using the Selector to click on intersections creates the exact point of intersection.) Overlay “Medium” segments to complete your ∆. 7. When given two ∆s, describe how to use the ASA test to determine if they are ≅ or not. 8. The SAA (or AAS) test can be “derived” from the ASA test. Using some basic relationships about ∠s in a ∆ and given that ASA is true, make an argument (informal proof) that SAA must also be true. Use pictures ad needed, just for illustration – no constructions. (New tab: 3d) d. Right-­‐hypotenuse-­‐Leg ∆ ≅ test (RHL) RHL Construction: “Copy” the right angle, one of the legs and the hypotenuse. 1. Use the steps below to construct a flexible right triangle, ∆ABC (keep the line segments “Medium”) a. Construct line that is approximately horizontal (NOT a line segment). Label the construction points A and B. b. Construct a ⊥ line through A. (Using the Construct menu.) c. Attach a point onto the ⊥ line and name it C. Construct segment BC . d. Hide the two ⊥ lines (but not points A & B), then construct segments AB and AC . Use the Marker tool to mark the right angle at A. 2. In a similar way as step 1, construct another pair of ⊥ lines off of ∆ABC and make then “Thin”. Construct the intersection and label it A’. Label the other Right angle is labeled A
construction point D (on the horizontal line). Move D out of the way (ou can use it later to help move the construction around.) 3. Construct a circle centered at A’ with radius ≅ to AB . Construct the intersection of that circle and line A'D and name it B’ (make the circle “Thin”). Overlay a “Medium” segment, A'B' . Orthocenter at B' with radius BC
4. Using line A'D (NOT segment A'B' ) construct an angle on line A'D at B’ that is ≅ to ∠B (Keep the rotated line “Thin.”) Make circle THIN
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5. Vertex C’ can only be in one possible location. Describe where to put C’ and why. Then construct and label C’ (Hint: Using the Selector to click on intersections creates the exact point of intersection.) Overlay “Medium” segments to complete your ∆. 6. When given two ∆s, describe how to use the RHL test to determine if they are ≅ or not. 7. Note that ASS is not a ∆ congruency test. However, RHL is a special case of ASS. Go to mathopenref.com – click on Congruence – click on Why SSA doesn’t work. Investigate and then write why SSA doesn’t always work but does work for the special case of RHL. (New tab: 3e) e. Angle-­‐Angle-­‐Angle ∆ test (AAA) [Note there is no “≅” symbol] AAA Construction: “Copy” two angles and the side between them. 1. Construct arbitrary ∆ABC (keep the line segments “Medium”). 2. Construct a Line off of ∆ABC and name one of its points A’ and label the other construction point D. (Make the line “Thin”). !###"
3. Construct an angle on line A'B' at A’ that is ≅ to ∠A. [Note: Move point D out of the way – you will only use D at the end to help move your construction around.] !###"
4. Attach an arbitrary point on line A'B' and name it B’. !###"
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5. Using the line A'B' (not the segment), construct an angle on line A'B' at B’ that is ≅ to ∠B. 6. Construct the intersection of the lines and name it C’. Overlay “Medium” segments to complete your ∆. 7. Move points A’, B’, and C’ around. How do the triangles compare? 8. Is AAA a congruency test? What kind of test is it?