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There are two right triangles, Triangle A and Triangle B. Triangle A has sides with lengths of 3, 4, and 5 and Triangle B has only two given side lengths: 3 and 4. The length of the hypotenuse is not given. Are these two triangles congruent?

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KizunaHighidn

RIGHT TRIANGLES

There are two right triangles, Triangle A and Triangle B. Triangle A has sides with lengths of 3, 4, and 5 and Triangle B has only two given side lengths: 3 and 4. The length of the hypotenuse is not given. Are these two triangles congruent?

The answer is they may be not congruent.

According to the Right Triangle Inequality Theorem, the sum of the two sides of a right triangle must be greater than the value of the third side.

Considering the names of the sides would be a,b and c, this would be the equations.

  • a + b > c
  • a + c > b
  • b + c > a

If the three equations satisfies the inequality, therefore it is a right triangle.

If you are going to test 3,4,5 as the sides, it is a right triangle.

Let a = 3, b = 4 and c = 5

  • a + b > c
  • 3 + 4 > 5
  • 7 > 5 (True)

  • a + c > b
  • 3 + 5 > 4
  • 8 > 4 (True)

  • b + c > a
  • 4 + 5 > 3
  • 9 > 3 (True)

Therefore it is a right triangle.

But how about if we change the value of c?

Let c = 6

  • a + b > c
  • 3 + 4 > 6
  • 7 > 6 (True)

  • a + c > b
  • 3 + 6 > 4
  • 9 > 4 (True)

  • b + c > a
  • 4 + 6 > 3
  • 10 > 3 (True)

Therefore, 6 can be the value of the hypothenuse and it forms a right triangle. The two triangles may not be congruent.

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