Questions and Answers
What is Thm 8-1 (Pythagorean Thm)?
If a triangle is a right triangle, the sum of the square of the lengths of the legs is equal to the square of the lengths of the hypotenuse (a^2 + b^2 = c^2).
What is a Pythagorean triple?
A set of non-zero whole numbers (a, b, and c) that satisfy the equation a^2 + b^2 = c^2.
Is 5, 12, 13 an example of a Pythagorean triple?
True
What is Thm 8-3?
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What is Thm 8-4?
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What is Thm 8-5 (45, 45, 90 triangle)?
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What is Thm 8-6 (30, 60, 90 triangle)?
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What does Soh Cah Toa represent?
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What do you do when you know that an angle in a right triangle = X and the side next to it equals 7 and the hypotenuse equals 10?
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Similar right triangles have equivalent ratios for their corresponding sides called _______________________
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What is the angle of elevation?
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What is the angle of depression?
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In order to solve a right triangle, you need at least ___ pieces of information.
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Study Notes
Pythagorean Theorem and Triples
- The Pythagorean Theorem states for right triangles, (a^2 + b^2 = c^2), where (c) is the hypotenuse.
- A Pythagorean triple consists of non-zero whole numbers (a), (b), and (c) that satisfy (a^2 + b^2 = c^2).
- The set (5, 12, 13) is an example of a Pythagorean triple.
Triangle Type Theorems
- Theorem 8-3: If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, the triangle is obtuse.
- Theorem 8-4: If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, the triangle is acute.
Specialized Triangle Theorems
- Theorem 8-5: In a 45-45-90 triangle, legs are congruent, and the hypotenuse is (\sqrt{2}) times a leg's length.
- Theorem 8-6: In a 30-60-90 triangle, the hypotenuse is twice the shorter leg, and the longer leg is (\sqrt{3}) times the shorter leg.
Triangle Properties
- Cutting an equilateral triangle with a bisector creates two scalene 30-60-90 triangles.
Trigonometric Functions
- Soh Cah Toa defines trigonometric ratios:
- Sine: (\sin = \frac{\text{Opposite}}{\text{Hypotenuse}})
- Cosine: (\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}})
- Tangent: (\tan = \frac{\text{Opposite}}{\text{Adjacent}})
Angle Calculations
- To find an angle (X) in a right triangle with a known adjacent side of 7 and hypotenuse of 10:
- Use: (\tan^{-1} \left(\frac{7}{10}\right)) to get approximately 35 degrees.
Similar Right Triangles
- Similar right triangles maintain equivalent ratios for their corresponding sides through trigonometric ratios.
Angles of Elevation and Depression
- The angle of elevation is the angle at which an observer looks upwards.
- The angle of depression is viewed when looking down at an object; these angles are equal due to alternate interior angles.
Triangle Solution Requirements
- To solve a right triangle, at least two pieces of information are needed, such as two side lengths or an angle with a side length.
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Description
Explore the properties and theorems of triangles with this quiz. Test your knowledge on the Pythagorean theorem, triangle type theorems, and specialized triangle characteristics. Perfect for understanding triangle fundamentals in geometry.
