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Midline Theorem and Related Theorems

Midline Theorem and Related Theorems. A. 4. 1. Y. F. X. 2. 3. B. C. Midline Theorem 1. In a triangle, the segment joining the midpoint of two sides is parallel o the third side and equal to one-half of it.

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Midline Theorem and Related Theorems

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  1. Midline Theorem and Related Theorems

  2. A 4 1 Y F X 2 3 B C Midline Theorem 1 In a triangle, the segment joining the midpoint of two sides is parallel o the third side and equal to one-half of it.

  3. Given: In ∆ABC, AX = XB, AY = YcProve: XYװBC and XY = 1/2BCConstruction: Produce XY to F so that XY = YF. Join FC. In ∆AXY,∆CFY Statements Reasons • XY = YF Construction • AY = YC Given • 1 =  2 Vertical ’s • ∆AXY  ∆CFY SAS • 3 = 4,AX = CF CPCTE • AXB װCF ALt. int. s equal • But AX = XB Given

  4. Statements Reasons • CF = XB Substitution • CF װXB XB is a part of AXB • XBCF is a parallelogram opp. sides equal and parallel • XF װBC, XF = BC Sides of parallelogram • XY = ½ XF Construction of XY=YF • XY = ½ BC Substitution • XY װBC XY is part of XF

  5. Midline Theorem 2 The segment through the midpoint of one side of a triangle to a second side bisects the third side.

  6. Median In a trapezoid, the segment joining the midpoints of the nonparallel sides is called the median of the trapezoid

  7. A B 3 1 X Y 2 4 C Z D Median of a Trapezoid In a trapezoid. the median is equal to one-half the sum of the lengths of the bases.

  8. Given: ABCD is a trapezoid with AB װDC AX = XD, BY = YCProve: XY = ½ (AB + DC) Construction: Join BX and produce it to meet CD produced at Z In ∆ABX, ∆DZX Statements Reasons • AX = XD Given • 1 = 2 Def. of vertical • 3 = 4 AIP • ∆ABX  ∆DZX SAA • ZX = XB, AB = DZ CPCTE

  9. In ∆BCZ Statements Reasons • BX = XZ Proved • BY = YC Given • XY = ½ CZ = ½ (CD+DZ) Midline Theorem • XY = ½ (CD+AB) Substitution

  10. t s A D m 2 1 5 E B n X C 3 4 6 F p Y Transversal Cut by Parallel Lines If three or more parallel lines cut off equal segments on one transversal, hey cut off equal segments on any transversal.

  11. Given: mװnװp, t and s are transversals such that AB = BCProve: DE = EFConstruction: Through D and E, draw DXװt and EYװt to cut BE and CF at X and Y respectively Statements Reason 1.AD װ BX, BE װ CY Segments of װ lines are parallel 2.DX AX, EY BC Construction 3.ADXB, BEYC are װgrams Def. of װgrams 4.AB = DX, BC = EY Opp. sides of װgram

  12. Statements Reasons 5.But AB = BC Given 6.DX = EY Substitution In ∆DXE, ∆EYF 7.1 =  2,  3=  4, Corr. s of 8. 2=  3 parallel lines 9.1=  4 Substitution 10.5=  6 Corr.  s of parallel lines 11.DX = EY Proved 12.∆DXE∆EYF SAA 13.DE = EF CPCTE

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