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Questions and Answers
What is the definition of Standard Position?
What is the definition of Standard Position?
- A right triangle whose hypotenuse is a radius of a circle of radius 1 (correct)
- A right triangle with one leg on the x-axis (correct)
- A triangle with all sides equal
- A triangle where all angles are 90 degrees
What does Theorem 9.1: Pythagorean Theorem state?
What does Theorem 9.1: Pythagorean Theorem state?
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 legs.
What is a Pythagorean Triple?
What is a Pythagorean Triple?
A set of three positive integers that satisfy the equation $a^2 + b^2 = c^2$.
List one common Pythagorean Triple.
List one common Pythagorean Triple.
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What does Theorem 9.2: Converse of the Pythagorean Theorem state?
What does Theorem 9.2: Converse of the Pythagorean Theorem state?
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What is Theorem 9.3: Pythagorean Inequalities Theorem about?
What is Theorem 9.3: Pythagorean Inequalities Theorem about?
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Study Notes
Standard Position
- A right triangle situated with its hypotenuse as a radius of a unit circle.
- Positioned such that one leg lies along the x-axis and the other leg is perpendicular to it.
Theorem 9.1: Pythagorean Theorem
- States that in any right triangle, the relationship between the lengths of the sides is defined by the equation ( a^2 + b^2 = c^2 ).
- In this equation, ( c ) represents the length of the hypotenuse.
Pythagorean Triple
- Refers to a group of three positive integers ( a, b, c ) that satisfy the equation ( a^2 + b^2 = c^2 ).
Common Pythagorean Triples and Some of their Multiples
- Examples of Pythagorean triples include:
- (3, 4, 5) and multiples like (6, 8, 10)
- (5, 12, 13) and multiples like (10, 24, 26)
- (8, 15, 17) and multiples like (16, 30, 34)
- (7, 24, 25) and multiples like (14, 48, 50)
Theorem 9.2: Converse of the Pythagorean Theorem
- Establishes that 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 classified as a right triangle.
Theorem 9.3: Pythagorean Inequalities Theorem
- For triangle ( \Delta ABC ) with ( c ) as the longest side:
- If ( a^2 + b^2 > c^2 ), the triangle is acute.
- Additional conditions apply when focusing on obtuse triangles.
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