Prove Converse of Pythagorean Theorem

Discussion in 'Algebra' started by nycmathguy, Jul 6, 2022.

  1. nycmathguy

    nycmathguy

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    College Algebra
    Chapter 1/Section 3

    Prove the converse of the Pythagorean Theorem. Explain each step along the way.
     
    nycmathguy, Jul 6, 2022
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  2. nycmathguy

    MathLover1

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    The converse of the Pythagorean Theorem is:

    If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

    That is, in ΔABC, if c^2=a^2+b^2 then ∠C is a right triangle, ΔPQR being the right angle.

    proof;
    We can prove this by contradiction.

    Let us assume that c^2=a^2+b^2 in ΔABC and the triangle is not a right triangle.

    Now consider another triangle ΔPQR. We construct ΔPQR so that PR=a, QR=b and ∠R is a right angle.

    [​IMG]

    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 and say:

    PQ=AB

    That is, all the three sides of ΔPQR are congruent to the three sides of ΔABC.

    So, the two triangles are congruent by the Side-Side-Side Congruence Property.

    Since ΔABC is congruent to ΔPQR and ΔPQR is a right triangle, ΔABC must also be a right triangle.

    This is a contradiction. Therefore, our assumption must be wrong.
     
    MathLover1, Jul 6, 2022
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    nycmathguy likes this.
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  3. nycmathguy

    nycmathguy

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    Great study notes. A job well-done!
     
    nycmathguy, Jul 6, 2022
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