Pregunta
If the square
of the length
of the longest
side of a
triangle is
equal to the
sum of the
squares of the Choose…
other two
sides, then
the triangle is
a right
triangle.
of the length
of the longest
side of a
triangle is
equal to the
sum of the
squares of the Choose…
other two
sides, then
the triangle is
a right
triangle.
Ask by Summers Fuentes. in the United States
Jan 24,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
If the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is a right triangle.
Solución
It looks like you’re referring to the converse of the Pythagorean theorem. Let’s complete and clarify your statement:
Complete Statement:
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.
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.
Explanation:
-
Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, if
is the hypotenuse and and are the other two sides, then: -
Converse of the Pythagorean Theorem: This states that if in a triangle, the square of one side equals the sum of the squares of the other two sides, then that triangle must be a right-angled triangle. In other words, if:then the triangle with sides
, , and is a right triangle with the right angle opposite side .
Implications:
-
Verification: This converse is often used to verify whether a given triangle is right-angled when the lengths of its sides are known.
-
Uniqueness: Only in right-angled triangles does this specific relationship between the sides hold true.
Example:
Suppose you have a triangle with sides of lengths 3 units, 4 units, and 5 units.
Suppose you have a triangle with sides of lengths 3 units, 4 units, and 5 units.
Check if it’s a right triangle:
Since
Conclusion:
Your statement is correct. The converse of the Pythagorean theorem provides a reliable method to determine whether a given triangle is right-angled based on the lengths of its sides.
Your statement is correct. The converse of the Pythagorean theorem provides a reliable method to determine whether a given triangle is right-angled based on the lengths of its sides.
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Simplifique esta solución Bonus Knowledge
Did you know that the relationship you just described is known as the Pythagorean theorem? It dates back to ancient Greece and is attributed to the philosopher Pythagoras. This theorem not only laid the foundation for geometry but also has far-reaching implications in various fields like physics and engineering!
When applying the Pythagorean theorem, always ensure you’re working with the correct sides – remember, the longest side in a right triangle is the hypotenuse. Common mistakes include misidentifying the hypotenuse or incorrectly calculating the square roots. Double-checking your math can save you from a lot of confusion!
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