4.5 the converse of the pathagorean theorem​

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4.5 the converse of the pathagorean theorem​

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Serenity 2 weeks 2021-09-09T11:50:21+00:00 2 Answers 0

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    2021-09-09T11:51:49+00:00

    If a^2 + b^2 = c^2, then the triangle is a right triangle.

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    2021-09-09T11:51:51+00:00

    Answer:

    The converse of the Pythagorean Theorem, {\bf (PQ)}^{\bf 2}={\bf a^2+b^2} is true

    Step-by-step explanation:

    The Pythagorean theorem or Pythagoras’s theorem is a statement about the sides of a right triangle.

    If the lengths of the legs are a and b, and the length of the hypotenuse is c, then, a^{2}+b^{2}=c^{2}

    To prove that the converse of the Pythagorean Theorem, (PQ)^2=a^2+b^2

    By the Pythagorean Theorem, (PQ)^2=a^2+b^2

    But we know that a^2+b^2=c^2 and  c=AB

    So,  (PQ)^2=a^2+b^2=(AB)^2

    That is, (PQ)^2=(AB)^2

    Since PQ and AB are lengths of sides, we can take positive square roots.

    PQ=AB

    That is, all the three sides of \triangle PQR are congruent to the three sides of \triangle ABC . So, the two triangles are congruent by the Side-Side-Side Congruence Property.

    Since \triangle ABC is congruent to \triangle PQR and \triangle PQR is a right triangle, \triangle ABC must also be a right triangle.

    This is a contradiction. Therefore, our assumption must be wrong.

    Therefore the converse of the Pythagorean Theorem,

    (PQ)^2=a^2+b^2

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