Triangle Midsegment Theorem and Perpendicular Bisectors

Triangle Midsegment Theorem

  • Definition: A midsegment of a triangle is a line segment connecting the midpoints of two sides of the triangle.

    • Example: In triangle ABC, if D and E are midpoints of sides AB and BC respectively, then DE is a midsegment.

  • Theorem: If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and its length is half the length of the third side.

    • If D and E are midpoints of AB and BC respectively, then DE∥ACDE \parallel AC and DE=12ACDE = \frac{1}{2}AC.

  • Identifying Parallel Segments:

    • If X, Y, and Z are midpoints of sides PQ, QR, and RS respectively, then:
      • XY∥QSXY \parallel QS
      • YZ∥PQYZ \parallel PQ
      • XZ∥PRXZ \parallel PR

Examples and Applications of Triangle Midsegment Theorem

  • Example 1:

    • Given triangle PQR with midpoints L, M, and N on sides PQ, QR, and RP, respectively.
    • Given: PR=46PR = 46, PQ=40PQ = 40, LV=17LV = 17.
      • Then, LM=12PR=12(46)=23LM = \frac{1}{2}PR = \frac{1}{2}(46) = 23
      • MN=12PQ=12(40)=20MN = \frac{1}{2}PQ = \frac{1}{2}(40) = 20
      • QR=2∗LV=2∗17=34QR = 2 * LV = 2 * 17 = 34
      • NR=12PR=1246=23NR = \frac{1}{2}PR = \frac{1}{2}46 = 23

  • Example 2:

    • Given that D, E, and F are midpoints of the sides of a triangle with CD=14CD = 14, EG=18EG = 18, and DF=24DF = 24.
      • Then, GE=18GE = 18
      • FH=18FH = 18
      • Perimeter of triangle CDE = CD+DE+EC=14+18+24=56CD + DE + EC = 14 + 18 + 24 = 56

  • Example 3:

    • Finding the value of x using the midsegment theorem.
      • If one side of a triangle is defined as 17x−1817x - 18 and the midsegment parallel to that side is defined as 7x7x, then:
        • 2∗7x=17x−182 * 7x = 17x - 18
        • 14x=17x−1814x = 17x - 18
        • 3x=183x = 18
        • x=6x = 6

Perpendicular Bisectors

  • Definition: A perpendicular bisector of a segment is a line or segment that intersects the given segment at a right angle and divides it into two congruent parts.

  • Perpendicular Bisector Theorem:

    • If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
      • If line AD is the perpendicular bisector of segment AB and AC equals CB, and angle ADB is a right angle, then AC = CB.

  • Converse of the Perpendicular Bisector Theorem:

    • If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.
      • If AC = CB, and AD = DB, then ED is perpendicular to AB.

Examples Involving Perpendicular Bisectors

  • Example 1:

    • Find the value of x, given that a point lies on the perpendicular bisector.
      • 7x+24=13x−487x + 24 = 13x - 48
      • −6x=−72-6x = -72
      • x=12x = 12

  • Example 2:

    • If 17x−9=9x+1317x - 9 = 9x + 13
      • 8x=228x = 22
      • x=228=114=2.75x = \frac{22}{8} = \frac{11}{4} = 2.75

  • Example 3:

    • Find RS, given expressions involving x.
      • 7x−2=4x+57x - 2 = 4x + 5
      • 3x=73x = 7
      • x=73x = \frac{7}{3}
      • If RS=4x+5RS = 4x + 5, then RS=4(73)+5RS = 4(\frac{7}{3}) + 5
      • RS=283+153=433RS = \frac{28}{3} + \frac{15}{3} = \frac{43}{3}