Day 66 Bellringer 1. Construct a perpendicular bisector to the given lines. a) b) HighSchoolMathTeachers@2018 Page 1
Day 66 Bellringer c) d) HighSchoolMathTeachers@2018 Page 2
Day 66 Bellringer 2. Identify two sides that are congruent in the triangle below. C 45 A 45 B HighSchoolMathTeachers@2018 Page 3
Day 66 Bellringer Answer Key Day 66 1. a) b) HighSchoolMathTeachers@2018 Page 4
Day 66 Bellringer c) d) 2. AC and AB HighSchoolMathTeachers@2018 Page 5
Day 66 Activity 1. Draw a line of length 2in and label it AB. 2. Position the compass at end A and extend it to end B, then draw a circle as shown below. A B HighSchoolMathTeachers@2018 Page 6
Day 66 Activity 3. Position the compass at the end B and using the same compass width draw a circle as shown A B HighSchoolMathTeachers@2018 Page 7
Day 66 Activity 4. Join the intersections of the circles with a straight line as shown. Label the point where the two lines meet as O. A O B HighSchoolMathTeachers@2018 Page 8
Day 66 Activity 5. Measure the length of AO. What do you get? 6. Measure the length of BO. What do you get? 7. Compare the two measurements in 5 and 6 above. HighSchoolMathTeachers@2018 Page 9
Day 66 Activity In this activity, students will construct a bisector to the line and measure the resulting portions to see if they are equal. Students will work in groups of at least three and each group is required to have a compass, a pencil, a ruler and a plain paper. Answer Keys Day 66: 1-4. No response 5. 2 in 6. 2 in 7. They are equal HighSchoolMathTeachers@2018 Page 10
Day 66 Practice Use the diagram below to answer questions 1 and 2. The point O is a bisector of the line QR. QO = 1.4in Q O R 1. What is the length of OR? 2. Find the length of QR Use the following information to answer questions 3 and 4. A point S is a bisector to the line AB. The length of AB is 12in. 3. What is the length of SB? 4. What is the length of AS? 5. At what ratio does point S divide line AB? Line RT divides another line UV in a ratio 1: 1. The two lines intersect at a point O. The length of OV is 2.4in. Use this information to answer questions 6 and 7. 6. What is the length of OU? 7. Find the length of the line UV. The line bisector of line CD intersects it at a point O. Line CD is 8in long. Use this information to answer questions 8-10. 8. What is the length of CO? HighSchoolMathTeachers@2018 Page 11
Day 66 Practice 9. What is the length of OD? 10. Write an equation relating OD and CD. 11. Write an equation relating OD and OC? Use the following information to answer questions 12-14 A man wanted to erect a security light post exactly in the middle of his rectangular plot. The plot measured 60ft by 80ft. In order to identify the middle of the plot he drew perpendicular bisectors to two adjacent sides and erected it at the point of the intersection of the two bisectors. 12. What was the shortest distance from the post to the short side of the plot? 13. What was the shortest distance from the post to the long side of the plot? 14. What was the distance from one vertex of the plot to the post? A point T divides line RK in the ratio 1:1. Line RK is 24in long. Use this information to answer questions 15 to 17. 15. Find the length of RT. 16. Find the length of TK. 17. Find the RT:TK HighSchoolMathTeachers@2018 Page 12
Day 66 Practice 18. Write the equation to relate RK and TK. A line AB bisects line ST through at a point Q. Line ST 13in long. Use this information to answer question 19 and 20. 19. What is the length of QT? 20. What is the length of QS? HighSchoolMathTeachers@2018 Page 13
Day 66 Practice Answer Keys Day 66: 1. 1.4in 2. 2.8in 3. 6in 4. 6in 5. 1: 1 6. 2.4in 7. 4.8in 8. 4in 9. 4in 10. CD = 2OD 11. OC = CD 12. 40 ft 13. 30 ft 14. 50ft 15. 12in 16. 12in 17. 2: 1 18. RK=2TK 19. 6.5in 20. 6.5in HighSchoolMathTeachers@2018 Page 14
Day 66 Exit Slip 1. Line PQ of length 10in is bisected by a point O. What is the sum of the lengths of the two portions? HighSchoolMathTeachers@2018 Page 15
Day 66 Exit Slip Answer Keys Day 66: 1. 10in HighSchoolMathTeachers@2018 Page 16
Day 67 Bellringer Consider KLM (not drawn to scale) shown below where NP = 2.3 in, LN = 3.1 in, NQ = 4.2 in and KQNP is a parallelogram. Use it to answer the following questions. M P 2.3 in N 4.2 in 3.1 in K Q L (a) Given that KQNP is a parallelogram find the length KP. (b) Hence find the length KM. (c) Find the length KQ on parallelogram KQNP (d) Hence find the length KL. (e) Compare the length of KL to that of NP. What do you notice? HighSchoolMathTeachers@2018 Page 17
Day 67 Bellringer Answer keys Day 67: (a) KP = 4.2 in. (b) KM = 8.4 in. (c) KQ = 2.3 in. (d) KL = 4.6 in (e) The length of NP is half that of KL HighSchoolMathTeachers@2018 Page 18
Day 67 Activity 1. On the plain paper, draw a triangle of suitable size and label it ΔKLM just like the triangle shown below. M 2. Measure the length of KM and a hence carefully locate point N, the midpoint of KM as shown below. K M L N K L 3. Similarly, measure the length of ML and a hence carefully locate point P, the midpoint of ML as shown below. M N P K L HighSchoolMathTeachers@2018 Page 19
Day 67 Activity 4. Join point N to point P using a ruler as shown below. M N P K L 5. Measure the lengths of NP and KL. What do you notice after comparing these lengths? 6. Now, join point P to point K using a broken line as shown below. M N P 7. Measure NPK and LKP then compare their measures. What main conclusion can be drawn about the relationship between NP and KL? K L HighSchoolMathTeachers@2018 Page 20
Day 67 Activity In this activity, students will discover the triangle midpoint theorem by drawing a line joining the midpoints of any two sides of a triangle. Students can work in groups of three or four. Each group should have a plain paper, a ruler and a protractor. Answer keys Day 67: 1. No response 2. Ensure that KN MN 3. Ensure that MP LP 4. No response 5. The length of NP is half that KL 6. No response 7. NP KL HighSchoolMathTeachers@2018 Page 21
Day 67 Practice Use the figure below to answer questions 1-8. In the figure, S is the midpoint of PQ, ST is parallel to QR, SU is parallel to PR, PST = 58 and PRQ = 58. P S T Q U R 1. Find the measure of SQU 2. Find the measure of QUS 3. Find the measure of PTS 4. Find the measure of TPS 5. Find the measure of QSU 6. Considering the angle measures you have found in questions. State whether ΔPST is congruent to ΔSQU or not. HighSchoolMathTeachers@2018 Page 22
Day 67 Practice 7. Given that TS RU, compare the length TS to the length RQ. 8. Considering your answer in question 7 above, is U is the midpoint of QR? Use the figure below to answer questions 9-12. In ΔKLM below, N, Q and P are the midpoints of KL, LM and KM respectively and LN QP. Show that LQPN is a parallelogram by completing the table below. K N P L Q M Statement Reasons LN QP 9. NP LQ 10. 11. QP 1 LK but LN 1 LK 2 2 LQ NP 12. HighSchoolMathTeachers@2018 Page 23
Day 67 Practice Use the figure below to answer questions 13-20. E and D are the midpoints of AB and AC respectively, ED DF and CF BE A 61 E 59 D F B C Given that E and D are the midpoints of AB and AC respectively, give two relationships between ED and BC in the table below? 13. 14. Fill in the table below. Measure of CDF 15. 16. Find the measures of the following pairs of angles: 17. CBE, hence BED Reason 18. BCF, hence CFD 19. Compare the pairs of angles in question 17 and 18. What do you notice? 20. What type of quadrilateral is most likely to be BCFE according to the angles you have found questions 17 and 18 above? HighSchoolMathTeachers@2018 Page 24
Day 67 Practice Answer keys Day 67: 1. 58 2. 58 3. 58 4. 64 5. 64 6. They are congruent 7. TS 1 RQ/ RQ 2TS 2 8. Yes Statement LN QP NP LQ Reasons 9. Triangle midpoint theorem 10. Triangle midpoint theorem 11. LN QP QP 1 2 LK but LN 1 2 LK LQ NP 12. NP 1 2 LM but LQ 1 2 LM 13. ED 1 BC / BC 2ED 2 14. ED BC Measure of CDF Reason 15. 59 16. Vertical angles / Opposite angles 17. CBE = 60, BED = 120 18. BCF = 120, CFD = 60 19. The angles in each pair are congruent 20. A parallelogram HighSchoolMathTeachers@2018 Page 25
Day 67 Exit Slip In the figure below TYUS is a parallelogram in ΔXYZ. YU is produced to point Z such that UY UZ and XS is also produced to point Z. Show that XS ZS by completing the table below the triangle. X T S Y U Z Statement TY SU Reason U is the midpoint of YZ if follows that S is also the midpoint of XZ Hence XS ZS HighSchoolMathTeachers@2018 Page 26
Day 67 Exit Slip Answer keys Day 67: Statements Reasons TY SU Opposite sides of a parallelogram are parallel U is the midpoint of YZ if follows that S is Triangle midpoint theorem also the midpoint of XZ Hence XS ZS HighSchoolMathTeachers@2018 Page 27
Day 68 Bellringer 1. Use the figure below to answer the questions that follow. Points S and T are the midpoints of sides AC and BC respectively. ST= 2.5in C S 127 T A B a) What is the size of ABC? b) What is the length of side AB? HighSchoolMathTeachers@2018 Page 28
Day 68 Bellringer 2. Use the figure below to answer the questions that follow. ABC is dilated to form CDE. B E C 1.5in 3in D A a) Find the scale factor of dilation. b) Where is the center of dilation? c) If side AC is 2.8in long, what is lethe ngth of CD? HighSchoolMathTeachers@2018 Page 29
Day 68 Bellringer Answer Key Day 68: 1. a) 53 b) 5in 2. a) 1 2 b) Point C c) 1.4in HighSchoolMathTeachers@2018 Page 30
Day 68 Activity 1. Plot a graph with a scale of 1square representing 1 unit as shown below y 4 2-6 -4-2 0 2 4 6 x -2-4 2. Draw a triangle with its vertices at points (0,0) B(5,1), C(1,4). 3. Identify the midpoints of sides AB and AC and label them as D and E respectively. 4. Join D and E with a straight line. What are the vertices of ADE? 5. Dilate ABC with a scale factor of 1 2. Does the triangle resulting from the dilation coincide with ADE? HighSchoolMathTeachers@2018 Page 31
Day 68 Activity In this activity, students will draw a triangle and a line passing through midpoints of two sides and compare the results with a dilation of the same triangle with a scale factor of 0.5. Students will work in groups of at least three and each group is required to have a graph paper, a pencil and a ruler. Answer Keys Day 68: 1-3. No response 4. A(0,0), D(2.5,0.5), E(0.5,2) 5. Yes HighSchoolMathTeachers@2018 Page 32
Day 68 Practice Use the diagram below to answer questions 1-5. AB is parallel to ST. SU = 8in, ST= 6in, and AB= 3in. (the figure is not drawn to scale) U A B S T 6 in 1. Which transformation will map STU onto ABU? 2. What is the length of AU? 3. What is the length of AS? 4. What is the length of BU? 5. What is the length of BT? Use the figure below to answer questions 6-10. AC RT and RT is twice the length of RT. C 3in R B 5in T 6. Which transformation will map ABC onto TBR? A HighSchoolMathTeachers@2018 Page 33
Day 68 Practice 7. What is the length of CR? 8. What is the length of BR? 9. What is the length of BT? 10. What is the length of AT? Use the figure below to answer questions 11 to 15. RQ is parallel to MN and NO=MO=10 in. O R 6in Q M 12in N 11. Which geometric transformation will map MNO onto QRO? 12. What is the length of RO? 13. What is the length of MR? 14. What is the length of OQ? 15. What is the length of NQ? HighSchoolMathTeachers@2018 Page 34
Day 68 Practice Use the figure below to answer questions 16-20. KJ = 1 BC. C 2 K A J B 16. Which geometric transformation will map ABC onto AJK? 17. Write an equation that relates AK and CK. 18. Write an equation that relates AK and AC. 19. Write an equation that relates AJ and BJ. 20. Write an equation that relates BJ and AB. HighSchoolMathTeachers@2018 Page 35
Day 68 Practice Answer Keys Day 68: 1. A dilation about point U with a scale factor of 1 2 2. 4in 3. 4in 4. 5in 5. 5in 6. A dilation about point B with a scale factor of 1 2 7. 1.5in 8. 1.5in 9. 2.5in 10. 2.5in 11. A dilation about point B with a scale factor of 1 2 12. 5in 13. 5in 14. 5in 15. 5in 16. A dilation about point C with a scale factor of 1 2 17. AK = CK 18. CK = 1 AC 2 19. AJ = BJ 20. BJ = 1 AB 2 HighSchoolMathTeachers@2018 Page 36
Day 68 Exit Slip 1. Points D and E are midpoints of AB and BC respectively. BDE is a dilation of ABC. What is the scale factor of dilation? C E A D B HighSchoolMathTeachers@2018 Page 37
Day 68 Exit Slip Answer Keys Day 68: 1. 1 2 HighSchoolMathTeachers@2018 Page 38
Day 69 Bellringer Use the following diagram to answer the questions S M T J A H F N K G Line AJ and FH are parallel. Line NG = 3.5 in and NG = 2MG. Angle GKH =116 and Angle AMS = 42. 1. Find the length of MN 2. Find the size of angle GTJ. 3. Find the size of angle GTJ. 4. Find the size of angle GNH. 5. Find the length of MN HighSchoolMathTeachers@2018 Page 39
Day 69 Bellringer Answer Keys Day 69: 1. 3.5 in 2. 116 3. 42 4. 42 5. 7 in HighSchoolMathTeachers@2018 Page 40
Day 69 Activity 1. Using 3 rods make a triangle of suitable size by connecting the rods with strings. 2. Label the vertices of the triangle as ABC and measure their length. 3. What are the dimensions of the triangle? 4. Pick any two sides and identify their midpoint using a ruler 5. Tie the last rod to form a line from one midpoint to the other. 6. Measure the distance between these midpoints HighSchoolMathTeachers@2018 Page 41
Day 69 Activity 7. Compare the distance in 6 above with the third side of the triangle ( the side whose midpoint was not identified in 4 above). 8. Find the shortest distance between the third side in 7 above and the line connecting the midpoints. 9. Is the shortest distance constant throughout the two rods? 10. What do you conclude based on 9 above? HighSchoolMathTeachers@2018 Page 42
Day 69 Activity In this activity, students will work in groups of at least 4. They will create a framework from metal, plastic or wooden thin rods and verify the midpoint theorem. Each group will require a 4 five metal, wooden or plastic rods or length 5 7 inches each, strings and a ruler. Answer Keys Day 69: 1. No response 2. The vertices are labeled 3. Answers vary 4-5.No response 6. Answer varies 7. The third side must be approximately twice line connecting the midpoint 8. No response 9. Yes 10. The Third side and the line connecting the midpoints are parallel HighSchoolMathTeachers@2018 Page 43
Day 69 Practice Use the following information to answer questions 1 8 In the figure below, N divides LQ in the ratio of 1:1 while PM = PM = MQ. A point T is on line PL such that PL = 2TL. TL = 3 in, LN = 5.5 in and PQ = 9 in. Angle PTM = 27. L N P 1. Which kind of lines are PL and MN? M Q 2. Find the relationship between the two lines above. 3. What is the measurement of MN. 4. What is the measurement of PL. 5. What is the measurement of MQ 6. What is the measurement of LQ. 7. What kind of lines are MN and TL if any. 8. Is there any linear relationship between the two lines in 7 above? 9. What kind of figure is TMNL? HighSchoolMathTeachers@2018 Page 44
Day 69 Practice 10. Explain your results above. 11. Find the size of angle LNM. 12. Find the side of angle MNQ 13. Find the size of angle PLN. Use the following diagram to answer the following questions. 14 20. In the diagram below, HK and D are midpoints of AG, GC, and CE respectively. G F E H K D A B C 14. Compare the length of HK and AC. 15. Compare the length of HK and AB. 16. Compare the length of KD and GE. 17. Provide the reason for your answer in 16 above. 18. If HKD is a straight line and K divides HD twice, what type of lines are HD and GE? 19. What is the linear relationship between GE and AC. 20. What kind of figure is KDEF. HighSchoolMathTeachers@2018 Page 45
Day 69 Practice Answer keys Day 69: 1. Parallel lines 2. PL = 2MN 3. 3 in 4. 6 in 5. 4.5 in 6. 11 in 7. They are parallel 8. MN = TL 9. Parallelogram 10. MN and TL are parallel while LN and TM are parallel too, Opposite sides are parallel 11. 153 12. 27 13. 27 14. HK = 1 2 AC 15. HK = AB 16. 2KD = GE 17. Due to mid-point theorem since K and D are midpoints of CE and CG respectively 18. HD = GE 19. GE = AC 20. Parallelogram HighSchoolMathTeachers@2018 Page 46
Day 69 Exit Slip In the figure below, R and E are the midpoints of YP and YH respectively. Find the size of angle REH. Y R E P 43 H HighSchoolMathTeachers@2018 Page 47
Day 69 Exit Slip Answer Keys Day 69 137 HighSchoolMathTeachers@2018 Page 48