Mid-Point Theorem

Mid-Point Theorem:

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of its length.

Converse of Mid-Point Theorem:

The straight line drawn through the midpoints of one side of a triangle, parallel to another side, bisects the third side.

Equal Intercept Theorem:

If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them also makes equal intercepts.

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  • Q1

    In the following figure, ABCD is a parallelogram. E and F are the midpoints of AB and CD respectively.
    If MP = 4 cm, find the length of PN.

    Marks:1
    Answer:

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  • Q2

    In the following figure, DE ll BC and D is the midpoint of AB. If AG = 3 cm, find the length of AF.

    Marks:1
    Answer:

    By converse of Midpoint Theorem, G is the midpoint of AF.

    Therefore, AF = 2AG = (2 x 3) cm = 6 cm.

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  • Q3

    In the following figure, ABCD is a square and E and F are the midpoints of AB and AD respectively. If EF = 3 cm, find the length of OA.

    Marks:1
    Answer:

    In square ABCD, E and F are the midpoints of AB and AD respectively. Therefore, by Midpoint Theorem, we get

    EF = (1/2) BD

    3 = (1/2) BD [As EF = 3 cm]

    BD = 6 cm

    Since diagonals of a square are equal and bisect each other,

    OA = (1/2) AC = (1/2) BD = 6/2 = 3 cm.

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  • Q4

    In the following figure, ABCD is a rectangle and G and H are the midpoints of BC and CD respectively. If GH = 4 cm, find the length of EF.

    Marks:1
    Answer:

    In triangle BCD, G and H are the midpoints of BC and CD respectively. By Midpoint Theorem, we get

    GH = (1/2) BD

    4 = (1/2) BD [As GH = 4 cm]

    BD = 8 cm

    In triangle ABD, E and F are the midpoints of AB and AD respectively. By Midpoint Theorem, we get

    EF = (1/2) BD

    = (1/2) 8 [As BD = 8 cm]

    = 4 cm.

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  • Q5

    In figure PQRS, PQ ll TU ll VW ll SR and
    RW = WU = UQ. If PT = 1.5 cm, find PS.

    Marks:1
    Answer:

    In Figure,
    PQ ll TU ll VW ll SR and RW = WU = UQ.
    Then, by equal intercept theorem SV = VT = PT
    Now, PS = PT + TV + VS = 3PT = 3 x 1.5 = 4.5 cm

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