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Math Help - Right Triangle Proof

  1. #1
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    Right Triangle Proof

    Triangle ABC is a right triangle with height h to the hypotenuse. The height divides the hypotenuse into segments a1 and b1. Prove that h^2 = a1 * b1 without using triangle similarity.
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  2. #2
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    Quote Originally Posted by MATNTRNG View Post
    Triangle ABC is a right triangle with height h to the hypotenuse. The height divides the hypotenuse into segments a1 and b1. Prove that h^2 = a1 * b1 without using triangle similarity.
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    Using Pythagoras' theorem,

    \left(a_1\right)^2+h^2=a^2

    a^2+b^2=\left(a_1+b_1\right)^2

    \left(b_1\right)^2+h^2=b^2

    Therefore

    \left(a_1\right)^2+h^2+\left(b_1\right)^2+h^2=\lef  t(a_1+b_1\right)^2

    from which the result follows
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  3. #3
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    If it is not known that triangle ABC is a right triangle but it is known that h^2 = a1 * b1, can it be proved that angle C must be a right angle? I know that the answer is yes. Any ideas as to how I would go about proving this?
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  4. #4
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    Quote Originally Posted by MATNTRNG View Post
    If it is not known that triangle ABC is a right triangle but it is known that h^2 = a1 * b1, can it be proved that angle C must be a right angle? I know that the answer is yes. Any ideas as to how I would go about proving this?
    Yes,

    as the non-right-angles are acute, we have

    a_1b_1=h^2\Rightarrow\frac{a_1b_1}{h}=h\Rightarrow  \frac{b_1}{h}=\frac{h}{a_1}

    These ratios are the tangents of two angles, which being equal causes the angles to be equal.
    Both inner triangles have a right-angle, therefore the third angles are also equal
    and the two acute angles sum to 90 degrees.
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  5. #5
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    right triangle proof

    Hi guys,

    Solution 0f followup question using Pythagorean Theorem and same side ids given h^2 =a1 x b1

    h^2 + a1^2 =a^2
    h^2 + b1^2 =b^2
    2h^2 + a1^2 + b1^2 =a^2 + b^2
    2a1xb1 + a1^2 + b1^2 = a^2 +b^2
    (a1 + b1 )^2 = a^2 + b^2
    triangle is a right triangle
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