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Geometry Chapter 5 Review Segment Perpendicular Bisectors are drawn perpendicular to the sides of the triangle passing through the side’s midpoint. Angle Bisectors are rays that cut the angle in half. The points on the angle bisector are equidistant to the sides of the angle. Medians are drawn from an angle to the midpoint of the opposite side of the triangle Name ____________________ Point of Concurrency Circumcenter Incenter Centroid Location in Triangle Associated Properties Acute: inside The circumcenter is Right: On equidistant to the Obtuse: Outside vertices. Always inside Always inside The incenter is equidistant to the 3 sides of the triangle Centroid is located two-thirds of the length of the median from the angle. AB 23 AC ; BC 13 AC “balance point” of the triangle Altitudes are drawn from an angle so that they are perpendicular to the line that contains the opposite side. Always, sometimes or never? 1) A midsegment of a triangle will ___________ be parallel to one side of the triangle. 2) The perimeter of the triangle formed by the midsegments of a triangle is _______ one half of the original triangle’s perimeter. 3) The medians of a triangle will __________ intersect outside of the triangle. 4) The perpendicular bisectors of a triangle will _______ intersect outside the figure. 5) An altitude of a triangle is ___________ perpendicular to a side of the triangle. 6) An angle bisector of a triangle goes from a vertex to the opposite side and cuts the side in half, __________. 7) The shortest side of a triangle is _________ opposite the largest angle. 8) 3 inches, 6 inches and 9 inches could ________ be the sides of a triangle. Define each term and draw a picture illustrating how it is located. Include all markings! 9) circumcenter: 10) What is its special property? 11) incenter: What is its special property? centroid: 12) midsegment: What is its special property? 13) Given: A FD is the perpendicular bisector of CE CF is the angle bisector of ACE Solve for the following values. a) y = ______ h) mFCB = ______ b) CD = ______ i) mCFB = ______ c) CE = ______ j) mFBA = ______ d) x = ______ k) mBFA = ______ e) mFDE = ______ l) f) mDCF = ______ m) mDFE = ______ F B 112 44 3x - 5 C 2y - 7 D E y-3 mBAF = ______ g) mFED = ______ 14) EM and LM are bisectors. 15) BD , CD , and AD are angle bisectors. A B 5 y F E 7 G L x 10 M y B z 9 A D x 6 E K x = ___________ Simplify the radical. y = ___________ x = _____ y = _____ z = _____ C 16) Point E is the centroid of ABC . Find the indicated lengths: DE ________ CF ________ EF ________ BD ________ CD ________ Perimeter DEB ________ EB ________ Perimeter EFC ________ A G 4 14 10 C 16 E D F 12 B 17) Determine whether or not each of the following sets of numbers can represent the lengths of sides of a triangle. Write “yes” or “no” and show why. a) 18) b) 1, 1, 3 c) 6, 6, 7, d) 12, 25, 13 Given two sides of a triangle, state the restrictions that apply to the 3rd side. Write this as an inequality using x as the third side. a) 19) 8, 17, 15 9, 10 b) 15, 25 c) x=_______ y=________ z=________ 1, 30, 20) d) 7, 7 If the perimeter of COP is 60 cm, what is the perimeter of BAD ? ____ O z x y 40 60 C 21) A B In RST, ST RT and RT RS. a) If one of the angles of the triangle is obtuse, which angle must it be? _____ b) If the measure of one of the angles of the triangle is 60, which angle must it be? _____ D Sketch the triangle below P 22) List the sides of the triangle in order from shortest to longest. N A 23) List the angles of the triangle in order from largest to smallest. x-1 T 9 B x+3 x 36 P E 12 15 P F O 24) List the 5 segments in the picture in order from shortest to longest. B A 96 R 25) 60 List the 5 segments in order from longest to shortest. Q 81 40 55 T D 72 80 26) Suppose a triangle has angles which measure 50, 60, and 70 and its sides have lengths of 3, 7, and 5. Draw a sketch of the triangle, labeling all sides and angles. 27) Complete each statement with , , or =. a) AB ____ DC b) mY _____ mW A B 40 45 25 A M N 20 Y C 68 c) MO _____ MN X 23 U 60 18 D Z W 58 O P 28) Solve for x in each problem. Use , , or = to set up the problem. a) b) 2x + 5 2x + 32 6 c) 54 50 50 9 9 68 66 6 29) 52 46 7x - 12 4x - 9 State whether each of the following inequality relationships is true or false. Given: m5 m6 AB = BD a) m5 m1 T F b) m3 mABC T F c) m2 m1 T F d) BC BA T F C 6 2 D 1 3 4 B 30) 2x + 8 5 A List the sides of STU in order from longest to shortest. mS x 2 , mT 5x 1 and mU 2x 3