Questions and Answers
What is the cosine ratio?
What is the geometric mean between two positive real numbers a and b?
x such that a/x = x/b
What is the projection of a point on a line?
The point where a perpendicular through the point to the line intersects the line.
What is the projection of a segment on a line?
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What is a proportion?
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What is a ratio?
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What defines similar polygons?
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What is the sine ratio?
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What is the tangent ratio?
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What is Postulate 15?
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What is Theorem 5-1?
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What is Theorem 5-2?
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What is Theorem 5-3?
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What is Theorem 5-4?
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What is Theorem 5-5?
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What is Theorem 5-6?
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What is Theorem 5-7?
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What is Theorem 5-8?
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What is Theorem 5-9?
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What is Theorem 5-10?
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What is Theorem 5-11?
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What is Theorem 5-12?
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What is Theorem 5-13?
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Study Notes
Trigonometric Ratios
- Cosine Ratio: In a right triangle, defined as the length of the side adjacent to an acute angle divided by the hypotenuse.
- Sine Ratio: In a right triangle, the ratio of the length of the side opposite an acute angle to the hypotenuse.
- Tangent Ratio: The ratio of the length of the side opposite an acute angle to the length of the side adjacent to that angle.
Means and Proportions
- Geometric Mean: The positive real number x such that the ratio a/x equals x/b for positive real numbers a and b.
- Proportion: An equation that asserts two ratios are equal.
- Ratio: Comparison of two numbers via division, where the quotient is termed the ratio.
Similar Polygons and Triangles
- Similar Polygons: Polygons with corresponding angles equal and corresponding sides in proportion.
- Postulate 15: Triangles are similar if all three angles are equal (AAA).
- Theorem 5-1: If two angles of one triangle equal two angles of another triangle, they are similar.
- Theorem 5-2: If two sides of one triangle are proportional to two sides of another and the included angles are equal, the triangles are similar.
- Theorem 5-3: Similarity is established if three sides of one triangle are proportional to three sides of another triangle.
Properties of Similarity
- Theorem 5-4: Similarity of polygons is reflexive, symmetric, and transitive.
- Theorem 5-5: For similar polygons, the ratio of their perimeters matches the ratio of corresponding sides.
Triangle Properties and Theorems
- Theorem 5-6: A line parallel to one side of a triangle divides the other two sides proportionally.
- Theorem 5-7: An angle bisector in a triangle divides the opposite side into segments proportional to the other two sides.
- Theorem 5-8: The lengths of corresponding altitudes in similar triangles share the same ratio as corresponding sides.
- Theorem 5-9: In a right triangle, drawing the altitude to the hypotenuse forms two triangles that are similar to each other and to the original triangle.
- Theorem 5-10: The Pythagorean Theorem states that the sum of the squares of the legs equals the square of the hypotenuse in a right triangle.
- Theorem 5-11: A triangle is a right triangle if the sum of the squares of two sides equals the square of the third side.
Special Right Triangles
- Theorem 5-12: In a 30°-60°-90° triangle, the hypotenuse is double the length of the short leg; the longer leg is √3 times the short leg.
- Theorem 5-13: In a 45°-45° right triangle, the hypotenuse measures √2 times the length of each leg.
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Description
Test your knowledge on similar polygons with this quiz covering important concepts such as cosine ratios and geometric means. Dive into the definitions and relationships that define these shapes in geometry. Perfect for students looking to understand the properties of polygons better!