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What is the Pythagorean Theorem?
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The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides: a² + b² = c².
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Who is credited with the discovery of the Pythagorean Theorem?
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The theorem is named after the ancient Greek mathematician Pythagoras.
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In the formula a² + b² = c², what does 'c' represent?
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In the formula, 'c' represents the length of the hypotenuse, which is the side opposite the right angle.
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In a triangle with sides of lengths 3, 4, and 5, is it a right triangle?
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Yes, because 3² + 4² = 9 + 16 = 25, which is 5², confirming it is a right triangle.
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What must be true for a set of three numbers to be a Pythagorean triple?
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A set of three whole numbers a, b, and c is a Pythagorean triple if a² + b² = c² holds true.
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Name one Pythagorean triple besides (3, 4, 5).
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One example is (5, 12, 13) since 5² + 12² = 13².
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How can the Pythagorean Theorem be applied in real life?
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It can be used to determine distances between points or to check the squareness of structures like buildings.
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If one leg of a right triangle is 9 and the hypotenuse is 15, what is the length of the other leg?
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Using the theorem: 9² + b² = 15², b² = 225 - 81 = 144, so b = 12.
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How does the Pythagorean Theorem relate to coordinate geometry?
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It’s used in the distance formula to find the distance between two points in a plane.
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Why doesn’t the Pythagorean Theorem apply to non-right triangles?
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Because the fundamental relationship a² + b² = c² only holds in right triangles.
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How is the Pythagorean Theorem useful in physics?
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It helps calculate resultant vectors, which can be used to find net forces or velocities.
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What does the Pythagorean Theorem say about the type of angle opposite the hypotenuse?
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The angle opposite the hypotenuse is a right angle (90 degrees).
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Can the Pythagorean Theorem be used in three dimensions?
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Yes, an extension called the 3D Pythagorean Theorem or space diagonal formula can be used.
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What is the converse of the Pythagorean Theorem?
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If a² + b² = c² is true for a triangle, then the triangle must be a right triangle.
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What type of triangle is formed by sides measuring 8, 15, and 17?
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It's a right triangle because 8² + 15² = 17².
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