How do you prove a triangle with the hypotenuse of

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ht1o4qgqdy

Answered question

2022-06-01

How do you prove a triangle with the hypotenuse of length 5 and other sides with lengths 3 and 4 is a right triangle?

Answer & Explanation

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prhljaju396r1

Beginner2022-06-02Added 3 answers

If a triangle has two perpendicular sides with lengths 3 and 4, the length of the remaining side is 5 by the Pythagorean theorem. Assume that a non-right triangle T with side lengths 3 , 4 , 5 exists. By the S S S criterion of congruence, it is possible to overlap such triangle with the previous right triangle, contradiction.
Unwrapped version: by S S S, there is a unique triangle with side lenghts 3 , 4 , 5, up to isometries. Since there is a right triangle with such side lengths, every triangle with side lenghts 3 , 4 , 5 is a right triangle.
Alternative, creative version: by Heron's formula, the area of a triangle with side lenghts 3 , 4 , 5 is 6. That implies the orthogonality of the sides with lenghts 3 and 4, since 6 = 3 4 2

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