Maths-
General
Easy

Question

Write a coordinate proof.
Any right isosceles triangle can be subdivided into a pair of congruent right isosceles triangles.(Hint: Draw the segment from the right angle to the midpoint of the hypotenuse.)

hintHint:

Prove using congruence criterion.

The correct answer is: any right isosceles triangle can be subdivided into a pair of congruent right isosceles triangles is proved.


    Complete step by step solution:
    Let ABD be the right triangle right angled at A. Here BD is the hypotenuse and
    C is the midpoint of the hypotenuse so that BC= DC.

    Here, AB = AD (since it is isosceles triangle)
    ∠DAB = 90°
    ∠ADB = ∠ABD = 450 (since the base angles of an isosceles triangles are equal)
    By the converse of the midpoint theorem, P is the midpoint of AD.
    ∠DPC = ∠DAB  = 90°(since PC ∥ AB)
    Consider  ⃤  DPC and  ⃤  APC,
    DP = AP (since P is the midpoint)
    ∠DPC = ∠APC  = 90°
    CP = CP (common side)
    ∴ ⃤  DPC and  ⃤  APC are congruent by SAS congruence criterion.
     ⇒ DC = AC (corresponding parts of congruent triangles)
    So ⃤  ACD is an isosceles triangle.
    ∠CDA = ∠DAC = 450 (since the base angles of an isosceles triangles are equal)
    ⇒∠ACD = 90°
    Likewise, AC = CB
    So ⃤  ACB is an isosceles triangle.
    ∠CBA = ∠CAB = 450 (since the base angles of an isosceles triangles are equal)
    ⇒∠ACDB= 90°
    Consider 2 triangles   ⃤  ACD   and  ⃤  ACB
    ∠ACD =∠ACB= 90°
    CA = CA (common side)
    CD=CB (C is the midpoint)
    Hence  ⃤  ACD   and  ⃤  ACB are congruent by SAS congruence criterion.
    Thus any right isosceles triangle can be subdivided into a pair of congruent right
    isosceles triangles is proved.

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