Math 3 Geogebra Discovery - Equidistance Decemeber 5, 2014

Transcription

1 Math 3 Geogebra Discovery - Equidistance Decemeber 5, 2014 Today you and your partner are going to explore two theorems: The Equidistance Theorem and the Perpendicular Bisector Characterization Theorem. Vocabulary: As you work through this activity, a few vocabulary words will be helpful. Equidistant - A point is equidistant from two points if the distances between the original point and the other two points are equal. In other words: A point is equidistant from two (or more) other points if the segments drawn between them are congruent. Example: Given ABC is isosceles with base BC, point A is equidistant from points B and C because AB = BC. A Distances are equal becuase segments are congruent B C Defining a Line - We have discussed this before, but to define a line you need two points. Defining a line means restricting its position to only one location. A single point does not define a line because it does not provide a direction. Example: With two points, the first point nails the line down the second defines the direction. Those two points define a specific location for the line, so they define the line With only one point, a line can swing in whatever direction it chooses!

2 The Equidistance Theorem Your first objective is to explore the equidistance theorem. Just like the case of the missing diagram you will draw the picture described in the given portion of the theorem. You will then determine what you can prove from there. Here s the theorem: The Equidistance Theorem If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. As we discussed in class, the if part of the theorem usually tells you what the givens are. In this case that would be:...a point lies on the perpendicular bisector of a segment... So, open a new window in geogebra. Save the file as EQT Construction - Your name Then carefully follow the directions below, answering the questions as you come upon them. Directions: 1. Use the segment tool to draw a segment near the middle of your screen. Name the endpoints of the segment A and B. Make both points (A and B) red. What are the characteristics of a perpendicular bisector? What does it do to a segment? 2. Since a perpendicular bisector, bisects the segment, use the midpoint tool (found in the drop down menu of the point tool) and select the segment AB to place the midpoint. Name this point M and make it blue. What two segments are now congruent because of the midpoint? Why? 3. Now we know a perpendicular bisector, is also perpendicular to the segment through the midpoint, so select the perpendicular tool and then select the point M and the segment AB. This should create a line through M, perpendicular to AB. How could we use GeoGebra to determine that the line is really perpendicular to AB?

3 4. Did you say: show that it makes right (or 90 ) angles? Let s check it out. Use the Angle tool and select AB and the line. An angle will appear, if the angle s measure is greater than 180, open object properties click on the angle and in basic choose, measure of angle between 0 and 180. Did you get a right angle now? Are you curious about whether the midpoint is actually a midpoint? How would you check that? 5. Did you think about measuring the segments? Good idea! Go to the drop down menu of the angle tool and you ll see a tool called length. Select the length tool and then click on first A, then M. A length should appear. Now click on first B, then M. A length should once again appear. Are the lengths the same? You can use the arrow tool to move the text around so it doesn t block your segment or line. 6. Now we can handle the second part of the givens (or I guess technically the first part...). Select the point tool and place a point on the perpendicular bisector, anywhere but at the midpoint of the segment. Name this point C and make it green. According to the theorem, C should be equidistant from the endpoints, A and B of AB. How can we determine if this is true? 7. As you have likely heard many times by now, the shortest distance between two points (in 3-dimensions) is a line. So one way to measure distance is to draw a segment between two points and measure it. Using the segment tool, click on A and then C to create AC, then click on B and then C to create BC. In the same way you measured AM and BM, measure AC and BC. What are the lengths of AC and BC? from A and B? Why? Does this indicate that C is equidistant 8. Use the arrow tool to drag point C up and down the perpendicular bisector. The lengths of AC and BC should change as you move C. When you move C do the lengths of AC and BC stay equal to each other?

4 9. Using your picture, fill in the blanks with the appropriate nouns. If a point ( ) lies on the perpendicular bisector of a segment ( ), then it ( ) is equidistant from the endpoints ( ) of the segment ( ). 10. Now looking at the picture below, decide which segments are congruent by the Equidistance Theorem (EQT). Fill in the blanks of the proof. A Given: EF is the perpendicular bisector of AB. E D C F 1. = 1. If a point ( ) lies on the perpendicular bisector ( ) of a segment ( ), then it ( ) is equidistant from the endpoints ( ) of the segment ( ). 2. = 2. If a point ( ) lies on the perpendicular bisector ( ) of a segment ( ), then it ( ) is equidistant from the endpoints ( ) of the segment ( ). 3. = 3. If a point ( ) lies on the perpendicular bisector ( ) of a segment ( ), then it ( ) is equidistant from the endpoints ( ) of the segment ( ). 4. = 4. If a point ( ) lies on the perpendicular bisector ( ) of a segment ( ), then it ( ) is equidistant from the endpoints ( ) of the segment ( ). 11. Note that you could have proved CB = CA by either the EQT or the definition of a bisector. What MUST you have in the Givens to use the EQT? B...and now the Perpendicular Bisector Characterization Theorem!!

5 Perpendicular Bisector Characterization Theorem This is the converse of the EQT. This can be a tricky converse to write, but give it a try! Did you get something like what s below? If you did, great job! It s tricky!! The Perpendicular Bisector Characterization Theorem - If two points are equidistant from the endpoints of a segment then they define the perpendicular bisector of that segment. Just like we did before we need to think about the given portion of the theorem....two points are equidistant from the endpoints of a segment... How do you tell if a point is equidistant from two other points? (If you can t just measure the distance with your handy ruler.) Now I want you to draw a few examples of the situation described above. Start out with a line segment, then the two equidistant points (congruent segments connect the point to the endpoints of the segment).

6 Below are the shapes we most often see with the Perpendicular Bisector Characterization Theorem (PBC). Remember that each shape can be rotated or reflected, so each picture could also be side-ways or upside down! C E B D The Kite The two equidistant points are on opposite sides of the segment. K J M L The Isosceles Triangle with Midpoint The two equidistant points are the vertex of the triangle and the midpoint of the base. F H I G The Tent The two equidistant points are on the same side of the segment.

7 Now, just like we did with the EQT, you will do a proof by construction of the PBC. Open a new window in GeoGebra and save it as PBC Construction - Your Name Directions 1. Despite the fact that you started with a segment with your hand sketches, you will start with circles here! Remember how all radii of a circle are congruent? It s a great way to create equidistance. So, use the circle with center at point and draw a circle. Name the center C and the point on the edge A. 2. Now, still using the same circle tool, pick a place anywhere but on the first circle (Circle C) to place the center. Then select the point A as the second point. Name the second circle s center D. 3. Use the intersect tool (found in the drop down menu of the point tool) to create a point at the second intersection of Circle D and Circle C. Name this point B. 4. Then, with the segment tool, draw segments AB, AD, AC, BD and BC. How do you know that C and D are equidistant from A and B? (Hint: How do you know AC = AD and BC = BD) 5. Use the line tool and select C and D as the points on the line. 6. Now use the intersection tool to draw a point at the intersection of CD and AB. Name this point M. The second part of the theorem says...the two points define the perpendicular bisector of the segment... So, the question is does CD perpendicularly bisect AB. How would you check this using GeoGebra? 7. Use the measurement tool to measure AM and BM. What did you find? What were the measures? What does this mean?

8 8. Use the angle tool and click on A, M, and C in that order. If the angle s measure is greater than 180, go into object properties and fix it. What is the measure of AMC? What does that mean? In this case, do C and D define the perpendicular bisector of AB? 9. As usual, we should try looking at a different version of the picture. While our construction is sound, and thus you did prove the theorem by construction, you have to be careful; sometimes preconceptions can mess with a construction. So, grab point D using the arrow tool. Move it around! Try making The Kite, The Isosceles Triangle and The Tent. As you move D around, what do you notice about the measures of AM and BM? What about the measure of AMC? 10. Using your picture, write the appropriate nouns on the blank lines in the theorem. If two points (, ) are equidistant from the endpoints (, ) of a segment ( ), then the two points (, ) define the perpendicular bisector ( ) of the segment ( ).

9 11. For each picture and givens below, decide which segment perpendicularly bisects which segment by the PBC and fill in the reason s blanks. A Given: DA = DB AC = BC D E C B 1. is the perpendicular bisector of 1. If two points (, ) are equidistant from the endpoints (, ) of a segment ( ), then the two points define the perpendicular bisector ( ) of the segment ( ). Q R Given: S is equidistant from Q and R T is equidistant from Q and R S T 1. perpendicularly bisects 1. If two points (, ) are equidistant from the endpoints (, ) of a segment ( ), then the two points define the perpendicular bisector ( ) of the segment ( ).

10 12. This one is a little proof. Z Given: Circle X ZW Y is isosceles with base ZY N X W Prove: W N ZY Y 1. Circle X 1. Given X Z Y 2. XZ = XY ZW Y is isosceles 3. Given W Y Z 4. W Z = W Y W N ZY 5. If two points (, ) are equidistant from the endpoints (, ) of a segment ( ), then the two points define the perpendicular bisector ( ) of the segment ( ). Notice the little symbols next to steps 2 and 4? They indicate the equidistance steps. It s a good way to check your work. The symbol can ONLY be written next to a congruent segment step or an equidistance step. The letter on the left is the point, the two letters on the right indicate to which points the original point is equidistant. The points on the right must be the same in both steps! If you still have homework time (be honest about it please), do problems 1, 2 and 3 on page 187 in the textbook.