Podcast
Questions and Answers
What is a ratio?
A comparison of 2 things.
What is a proportion?
2 equal ratios
In the proportion a/b=c/d, which are the means?
B and C
In the proportion a/b=c/d, which are the extremes?
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1:2:3 is an _____ _____.
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Can extended ratios be written as fractions?
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What do a/b=c/d, b/a=d/c, a/c=b/d, and a+b/b=c+d/d all multiply out to?
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What does 'similar' mean in geometry?
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Can similar polygons be congruent?
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What are the angles of similar polygons?
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What are the sides of similar polygons?
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What is a scale factor in geometry?
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What is the arithmetic mean?
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What is the geometric mean?
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If you have a triangle with its 3 angles in a ratio of 1:2:3, what would your equation be to find each of the angles? (Use x)
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What does AA~ mean in triangle similarity?
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What does SSS~ indicate?
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What does SAS~ signify?
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What does CPSTP stand for?
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What does the Side Splitter Theorem state?
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What does the Triangle Angle Bisector Theorem state?
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Study Notes
Ratios and Proportions
- Ratios compare two quantities, illustrating their relative sizes.
- A proportion represents two equal ratios, indicating a relationship between four quantities.
- In the proportion a/b = c/d, B and C are called the means, while A and D are referred to as the extremes.
Extended Ratios
- An extended ratio, such as 1:2:3, compares three or more quantities.
- Extended ratios cannot be represented as simple fractions.
Properties of Proportions
- Proportions lead to the equality of products: Bc = ad for the established ratios.
- Similar polygons maintain angles that are equal and sides that correspond proportionally.
Similar Polygons
- Similar polygons may be congruent, meaning they could have the same shape and size.
- The sides of similar polygons must be in proportion, and the angles are equal.
Scale Factor
- The scale factor illustrates the ratio of corresponding sides between similar figures.
Means of Calculation
- The arithmetic mean of a set of numbers is found by summing the values and dividing by their count.
- The geometric mean requires multiplying the numbers and taking the square root.
Triangle Angle Relationships
- For a triangle with angles in the ratio 1:2:3, the equation representing the angles is x + 2x + 3x = 180.
- Triangles are similar if two pairs of angles are congruent (AA similarity).
- The SSS similarity criterion states that if the corresponding sides of two triangles are proportional, then those triangles are similar.
- SAS similarity occurs when two sets of sides are proportional and two angles are congruent in two triangles.
Important Theorems
- CPSTP (Corresponding Parts of Similar Triangles are in Proportion) underlines the proportional relationship of corresponding parts in similar triangles.
- The Side Splitter Theorem indicates that a line parallel to one side of a triangle divides the other two sides proportionally.
- The Triangle Angle Bisector Theorem establishes that an angle bisector divides the opposite side in proportion to the other two sides of the triangle.
Studying That Suits You
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Description
Test your knowledge of key concepts in Geometry Chapter 7 with these flashcards. This quiz covers important terms such as ratios, proportions, and the concepts of means and extremes. Dive in and reinforce your understanding of these foundational geometric principles.
