Module 8 Special Segments in Triangles: Review Date: 8.3 Coordinate Geometry

1) The coordinates of the vertices of a triangle are 𝐸 (1,2), 𝐹(5,6), 𝑎𝑛𝑑 𝐺(3, −2) a) Find coordinates of 𝐻(midpoint of 𝐸𝐺̅̅ ̅̅ ) and 𝐽 (midpoint of 𝐹𝐺̅̅ ̅̅ ) b) Verify that 𝐻𝐽 ||𝐸𝐹 c) Verify that 𝐻𝐽 = 1

2 𝐸𝐹

2) The following coordinates are the midpoints of the sides of a triangle. Find the coordinates of the vertices of the triangle. 𝐴(2,1), 𝐵(−1, − 2), 𝐶(3, − 3)

For each, graph, write 2 equations for 2 special segments, and find point(s) of concurrency.

3) 𝐴(−2, −2), 𝐵(2,3), 𝐶(2, −2) Find orthocenter and circumcenter

4) 𝐴(−1,2), 𝐵(3,6), 𝐶(4,2) Find centroid (solve the system)

5) 𝐴(−2, 1), 𝐵(0, −2), 𝐶(5, −2) Find centroid and orthocenter (solve the system)

6) 𝐴(−3, 2), 𝐵(3,3), 𝐶(−1, −2) Find circumcenter (solve system- both x and y coordinates are pretty rough)

7)𝐴(0,2) 𝐵(0,0) 𝐶(2,0) Find orthocenter and centroid (solve the system)

8) 𝑋(1, −2), 𝑌(−5, −2), 𝑍(−1,6) Find the circumcenter

8.2 Algebra

9)Find the value of 𝑥 if perimeter of ∆𝐴𝐵𝐶 = 30

10) Point S is the centroid of 'RTW, RS = 4, VW = 6, and TV= 9. Find the length of each segment. a) SU = __________ b) RU = __________ c) TS = __________ d) SV = __________

11) BD, EC, and AF are medians. Find x in each. Leave your answers as fractions a) 𝐴𝐺 = 3𝑥 − 1 and 𝐴𝐹 = 3

4 𝑥 + 2 b) 𝐸𝐺 = 1

4 𝑥 + 5 and 𝐺𝐶 = 5𝑥−24

c) 𝐺𝐷 = 3𝑥+2

5 and 𝐵𝐷 = 6𝑥

12) Find AB and CV

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13) 𝑄𝑂, 𝑂𝑅, 𝑎𝑛𝑑 𝑆𝑂 are perpendicular bisectors. Find: a) QO= b) MP= c) MO= d)PR= e) If 𝑂𝑃 = 2𝑥 + 9, find x. f) OR=

14) Point T is the incenter of 'PQR, 𝑚∠𝑃𝑅𝑇 = 24° Find: a) 𝑚∠𝑊𝑅𝑄 b) If 𝑇𝑈 = (2𝑥 – 1), find x c) ST d) SR e) PR f) 𝑚∠𝑊𝑃𝑆

8.1 Vocabulary and Constructions 17) Sketch and label the 4 points of concurrency and the segments that create them (give names of both) a)

b)

c) d)

18) Construct the orthocenter of this triangle 19) Construct the incenter of this triangle (Include appropriate circle)

20) Construct the centroid 21) Construct the circumcenter (Include appropriate circle)

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