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13 September, 07:03

Sketch and prove the following for the two congruent triangles ∆ABC and ∆DEF.

The medians drawn from vertex A and D are congruent.

Label medians MA and ND; then △ABM≅△

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  1. 13 September, 08:31
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    See explanation

    Step-by-step explanation:

    Given: ΔABC ≅ ΔDEF

    AM, DN - medians

    Prove: AM ≅ DN

    Proof:

    1. Congruent triangles ABC and DEF have congruent corresponding parts:

    AB ≅ DE; BC ≅ EF; ∠ABC ≅ ∠DEF.

    2. BM ≅ MC - definition of the median AM;

    3. EN ≅ NF - definition of the median DN;

    4. AB ≅ 2BM, EF ≅ 2EN

    BM ≅ 1/2 AB,

    EN ≅ 1/2 EF,

    thus, BM ≅ EN

    5. Consider two triangles ABM and DEN. In these triangles /:

    AB ≅ DE (see 1)); BM ≅ EN (see 4)); ∠ABM ≅ ∠DEN (see 1).

    So, ΔABM ≅ ΔDEN by SAS postulate.

    6. Congruent triangles ABM and DEN have congruent corresponding sides BM and DN.
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