Unit 6 Guided Notes. Task: To discover the relationship between the length of the mid-segment and the length of the third side of the triangle.

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1 Unit 6 Guided Notes Geometry Name: Period: Task: To discover the relationship between the length of the mid-segment and the length of the third side of the triangle. Materials: This paper, compass, ruler Steps: 1. Using a straightedge, construct a triangle. Do this next to the image of a triangle above, but don t make your triangle congruent to it. 2. Using a compass, bisect each side of the triangle to locate the midpoint of each side. 3. Connect the midpoints to form the three midsegments. 4. Measure the midsegments and the third sides for each pairing. Record the lengths. 5. Record the data for your triangle. Compare your results with your group and make a conjecture regarding the relationship between the length of the midsegment and the length of the third side of the triangle. Data: Midsegment Third Side DE = C = EF = C = FD = = Conjecture: We think that a midsegment and the third side of a triangle. The Triangle Midsegment Theorem: If a segment joins the midpoints of two sides of a triangle, then Practice 1. Name the triangle side that is parallel to the given segment. a. XY b. XZ c. ZY 2. Points M, N, and P are the midpoints of the sides of QRS. QR = 30, RS = 30, and SQ = Find MQ. 4. Find MP. 5. Find PN

2 Proofs Given: UX XW, WY =!! WV Prove: XY is a midsegment of UWV. Statements Reasons Given: XY is a midsegment of UWV. Prove: WXY XUV Statements Reasons

3 The Perpendicular isector Theorem: If a point is on the perpendicular bisector of a segment, then Sketch: The Converse of the Perpendicular isector Theorem is also true: If a point is equidistant from the endpoints of a segment, then Sketch: Practice Find x, then find IK. Draw the perpendicular bisector of. Proofs Given: WXYZ is a rhombus. Prove: WVY ZVX Statements Reasons

4 ngle isector Theorem: If a point is on the bisector of an angle, then Sketch: The Converse of the ngle isector Theorem is also true: If a point in the interior of an angle is equidistant from the sides of an angle, then Sketch: Practice How is MR related to PR? How do you know? Find MR and PR. Find the value of the variable. Then, identify which theorem you used to write the equation. x = x = Theorem used: Theorem used: x = y = Theorem used: Theorem used:

5 Proofs Given: PM OP, MN ON, PM MN m POM = x + 17, m NOM = 3x 5 Prove: x = 11 Statements Reasons Points of Concurrency When three or more lines intersect at one point, they are. n of a triangle is a segment from a vertex of the triangle to the line containing the opposite side. Concurrency of ltitudes Theorem: The lines that contain the of a triangle are concurrent. This point of concurrency is called the. Sketches of altitudes: The orthocenter can be inside, outside, or on the triangle.

6 of a triangle is a segment whose endpoints are a and the of the opposite side. Concurrency of Medians Theorem: The of a triangle are concurrent at a point that is the distance from each vertex to the midpoint of the opposite side. This point of concurrency is called the. Point G is the centroid of C, D = 8, G = 10, E = 10, C = 16 and CD = 18. Find the length of each segment. D = CG = GE = C = E = = GD = F = Concurrency of Perpendicular isectors Theorem: The of the sides of a triangle are concurrent at a point equidistant from the vertices. This point of concurrency is called the. The circumcenter can be inside, outside, or on the triangle.

7 In the diagram, point O is the circumcenter. Find the indicated measure. MO = MN = PR = SP = m MQO = If OP = 2x, find x. x = Concurrency of ngle isectors Theorem: The of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle. This point of concurrency is called the. In an equilateral triangle, all four points of concurrency are the same point! Practice Is segment a midsegment, perpendicular bisector, angle bisector, median, altitude, or none of these?

8 In the diagram, point G is the circumcenter. Find the indicated measure. G = CF = FC = G = = GF = m DG = IF G = (2x 15), find x. x = Point T is the incenter of ΔPQR. ST = If TU = (2x 1), find x. x = If m PRT = 24º, then m QRT = If m RPQ = 62º, then m RPT = Point S is the centroid of ΔRTW, RS = 4, VW = 6, and TV= 9. Find the length of each segment. RV = RU = TS = SU = RW = SV = Point G is the centroid of C. Use the given information to find the value of the variable. If FG = x + 8 and G = 6x 4, x = If CG = 3y + 7 and CE = 6y, y =

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