Congruent triangles are always similar. Which of the following statements is an example of the statement above? Select all that apply.

Transcription

1 Name Triangles Part 2 Triangle Similarity Part 1 Independent Practice 1. Consider the statement below: Date Page 1 Congruent triangles are always similar. Which of the following statements is an example of the statement above? Select all that apply. ngles are the same, but sides are proportional to each other. Sides are the same size. dilation of a scale factor 1. Corresponding angles and corresponding sides are congruent. dilation of a scale factor of Determine if the two triangles are similar. If so, write a similarity statement for the triangles. Justify your answer. B C D E Statement Reason &' &( = 3. &+ &, = 4. BD~ CE 4. Section 7 Topic 1

2 3. re the following triangles similar? Justify your answer. Page 2 M 12 C 14 6 D O N L Consider the following figure and proof. G E Given: G TE Prove: GR~ TRE R T Statement Reason Given lternate Interior ngles are Congruent lternate Interior ngles are Congruent 4. GR~ TRE 4. Section 7 Topic 1

3 Page 3 5. Before rock climbing, Fernando, who s 5.5 ft. tall, wants to know how high he will climb. He places a mirror on the ground and walks six feet backwards until he can see the top of the cliff in the mirror. Determine the similarity theorem or postulate that you can use to determine the height of the cliff. 6. Determine what similarity postulate or theorem we can use to determine the value of x, the width of the river. x ft. 90 ft. 120 ft 135 ft. Section 7 Topic 1

4 7. Consider the following figure and proof. G Page 4 E Given: RE = 2R and RT = 2GR Prove: GR~ TER R T Statement Reason Given Vertical ngles Proportionality of Sides 4. GR~ TER 4. Section 7 Topic 1

5 Name Triangles Part 2 Triangle Similarity Part 2 Independent Practice Date Page 5 1. Before rock climbing, Fernando, who is 5.5 ft. tall, wants to know how high he will climb. He places a mirror on the ground and walks six feet backwards until he can see the top of the cliff in the mirror. If the mirror is 34 feet from the cliff side, determine the height of the cliff. 2. Determine the width of the pond. x ft. 90 ft. 120 ft 135 ft. Section 7 Topic 2

6 Page 6 3. The following triangles are not similar. Determine the ratio between MCD and OLN. How could you change change the measurement(s) to make them similar? M 12 C 14 6 D O N L Basketball star Mumford (a six foot senior forward) places a mirror on the ground x ft. from the base of a basketball goal. He walks backward four feet until he can see the top of the goal, which he knows is 10 feet tall. Determine the how far the mirror is from the basketball goal. Justify your answer m tall child is standing next to a flagpole. The child s shadow is 1.2 m long. t the same time, the shadow of the flagpole is 7.5 m long. How tall is the flagpole? Section 7 Topic 2

7 6. Consider the following figure and proof. Given: RM SN, RM MS, SN NT, Prove: RSM~ STN T S N R M Page 7 Statement Reason Section 7 Topic 2

8 Name Triangles Part 2 Triangle Midsegment Theorem Part 1 Independent Practice Date Page 8 1. Consider SR. Write the three pairs of parallel segments in SR. N S WE RS NE S W E WN R R 2. In SR, W, E, and N are midpoints. Determine the lengths of each segment using the number bank Part : Part B: S = WE = 50 W 30 E Part C: Part D: EN = R + RS S = R N 40 S Section 7 Topic 3

9 3. In the diagram below, R is located at 24, 0, N is located at 12, 18, T is located at 12, 6, and E is located at 18, 15. ssume that N, K, T, E, I, and B are midpoints. Page 9 N E S K B T I R Complete the following table. Vertices S K I B Coordinates 4. s an answer to a test, Monica sees the figure below and determines that IR E. Monica s teacher counted it as incorrect. Determine the error in Monica s reasoning. I N R E Section 7 Topic 3

10 Page Identify three pairs of parallel segments in each diagram and write the pairs in the blanks. WK SN WR NE 12 E 10 RK ES 12 W K 10 N 7 R 7 S 6. Determine the value of x. x 81 Section 7 Topic 3

11 7. Consider the following figure and proof. K Page 11 J Given: bisects JK, C bisects KL, B bisects JL, Prove: JKL~ CB C B L Statement Reason Section 7 Topic 3

12 Name Triangles Part 2 Triangle Midsegment Theorem Part 2 Independent Practice Date Page In SMD, point B is the midpoint of SM, point N is the midpoint of MD, and point T is the midpoint of DS. 4x B 98.3 M 102. S T 2 5 y N 7 10 z D Part : Find the length of SM. Part B: Determine the value of x + y + z. Part C: Find the value of NT + TB BN. Section 7 Topic 4

13 2. Determine the value of x. Page 13 4x x 3. Consider the following figure. 2h + 1 D M N h k T 3h 6 B S Part : Determine the value of h. Part B: Determine the value of k. Section 7 Topic 4

14 4. Determine the length across the river, x, to the nearest hundredth. Page 14 x ft ft. 5. The coordinates of the vertices of a triangle are M 4, 1, 3, 3, and N 2, 3. Part : Determine the coordinate of J, the midpoint of M. Part B: Determine the coordinate of L, the midpoint of N. Part C: Prove that JL MN. Section 7 Topic 4

15 Name Triangles Part 2 Triangle Inequalities Independent Practice Date Page Consider the following triangle side lengths and determine if the triangle could exist. Justify your answer. Part : 21, 18, 17 Part B: 3, 12, 8 2. Consider the following figure. H C K 73 M H Determine which of the following statements must be true. H < CH B HM > CH C M = CK D MH < HC Section 7 Topic 5

16 3. Determine the range of possible values of x. Page 16 H 6.33 Y M 4. Find the range of possible values of x. (3x 7) Section 7 Topic 5

17 5. Consider the following figure and proof. Given: is the midpoint of ET, m CH = m HC, m EC > m TH C Page 17 E Prove: CE > HT Complete the two-column proof below H T Statement Reason 1. m CH = m HC 1. Given 2. C = H is the midpoint of ET 3. Given 4. E T Congruent segments have equal length 6. m EC > m TH 6. Given Section 7 Topic 5

18 Name Triangles Part 2 Triangle Inequalities Independent Practice 1. Consider the figures below. D Date I Page 18 E W R F bony used the above diagram to conclude that DW IF. Explain the rationale behind bony s conclusion and justify whether or not her conclusion is correct. 2. Consider the following figure. H C K 73 M Q Jazeel used the above diagram to conclude that HM CQ. Explain the rationale behind Jazeel s conclusion and justify whether or not his conclusion is correct. Section 7 Topic 6

19 3. Determine the range of possible values of x. H Page 19 Given: M YM; HM bisects MY Prove: Y Y Based on the above figure and the information below, complete the following two-column proof. M Statements Reasons 1. M YM HM bisects MY MH YMH HM HM HM YHM Y 6. Section 7 Topic 6

20 4. Consider the figure below. Page 20 I (2a, 2c) L M B (0, 0) P (2a, 0) The above figure shows BIP where L is the midpoint of BI and M is the midpoint of IP. Part : Prove that BIP~ LIM. Part B: Prove algebraically that the area of LIM is one-fourth the area of BIP. Part C: Justify whether or not CPCTC can be used in a scenario like the one presented in the above figure. Section 7 Topic 6

21 5. Consider the figure below. Page 21 Y C N Part : Based on the above figure and a given statement, Cassidy was able to conclude that NYC PC because of SS. Determine what is the given statement. P Part B: Prove that NY P both theoretically and applying transformations. Section 7 Topic 6

22 Name Date Page 22 Triangles Part 2 Inscribed and Circumscribed Circles of Triangles Independent Practice 1. Consider the figure below. line n B line m P C line p Dante argues that point P is the circumcenter of the triangle. Determine whether Dante is correct and justify your answer. 2. Consider the following figure and prove that the three angle bisectors of the internal angles of CMI are concurrent in point P. M P I C Section 7 Topic 7

23 3. Ms. Calcutta, the owner of a business park is adding a recycling station for every three office buildings. To make it easier for the tenants, she is going to to use the circumcenter of the three office buildings to place the recycling stations. By doing this, it will be easier to get to the station, because it is equidistant from the three office buildings. Part ; Justify the rationale behind Ms. Calcutta s decision to use the circumcenter. Page 23 Part B: There is a garbage dumpster exactly at the midpoint of each pair of office buildings. Suppose that there are sidewalks connecting each office building to each other and to each recycling station. What is the relationship between the building-to-building sidewalk and the station-to-dumpster sidewalk? 4. Consider the figure below and mark the line segments that are congruent. Section 7 Topic 7

24 Name Triangles Part 2 Medians in a Triangle Independent Practice 1. Consider the figure below. Date Page 24 B U I T R O S IR, US, and OB are all medians of BRS, and T is the centroid. IR = 10.8, BT = 4.5, UT = Find RT, TI, OB, and US. 2. Describe the similarities and differences between the circumcenter of a triangle and the centroid of a triangle. Section 7 Topic 8

25 Page Describe the similarities and differences between the incenter of a triangle and the centroid of a triangle. 4. Consider the triangle below. I S C L D Y Prove that c is the centroid of IY. Section 7 Topic 8