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Questions and Answers
Two rectangles are similar. One rectangle has a length of 8 cm and a width of 5 cm. The other rectangle has a length of 12 cm. What is the width of the second rectangle?
Two rectangles are similar. One rectangle has a length of 8 cm and a width of 5 cm. The other rectangle has a length of 12 cm. What is the width of the second rectangle?
- 7.5 cm (correct)
- 6 cm
- 10 cm
- 9 cm
If two triangles are similar, which of the following statements is NOT always true?
If two triangles are similar, which of the following statements is NOT always true?
- The triangles have the same shape.
- The corresponding sides are proportional.
- The corresponding angles are congruent.
- The triangles have the same area. (correct)
A tree casts a shadow that is 15 meters long. At the same time, a 2-meter tall person casts a shadow that is 3 meters long. What is the height of the tree?
A tree casts a shadow that is 15 meters long. At the same time, a 2-meter tall person casts a shadow that is 3 meters long. What is the height of the tree?
- 10 meters (correct)
- 6 meters
- 12 meters
- 8 meters
A scale drawing of a rectangular room is 5 cm long by 3 cm wide. The actual room is 10 meters long. What is the actual width of the room?
A scale drawing of a rectangular room is 5 cm long by 3 cm wide. The actual room is 10 meters long. What is the actual width of the room?
Triangle ABC is similar to triangle DEF. If AB = 6 cm, BC = 8 cm, and DE = 9 cm, what is the length of EF?
Triangle ABC is similar to triangle DEF. If AB = 6 cm, BC = 8 cm, and DE = 9 cm, what is the length of EF?
A map has a scale of 1:25,000. If two towns are 5 cm apart on the map, what is the actual distance between the towns?
A map has a scale of 1:25,000. If two towns are 5 cm apart on the map, what is the actual distance between the towns?
Which of the following is NOT a criterion for proving two triangles similar?
Which of the following is NOT a criterion for proving two triangles similar?
Two similar triangles have a scale factor of 3:4. If the perimeter of the smaller triangle is 12 cm, what is the perimeter of the larger triangle?
Two similar triangles have a scale factor of 3:4. If the perimeter of the smaller triangle is 12 cm, what is the perimeter of the larger triangle?
Flashcards
Similar Figures
Similar Figures
Figures with the same shape but not necessarily the same size.
Congruent Angles
Congruent Angles
Corresponding angles in similar figures are equal.
Proportional Sides
Proportional Sides
The sides of similar figures have a constant ratio.
Scale Factor
Scale Factor
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Polygons Similarity
Polygons Similarity
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AA Similarity Postulate
AA Similarity Postulate
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SSS Similarity Theorem
SSS Similarity Theorem
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SAS Similarity Theorem
SAS Similarity Theorem
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Study Notes
Definition and Characteristics
- Similar figures are figures that have the same shape but not necessarily the same size.
- Corresponding angles in similar figures are congruent (equal).
- Corresponding sides in similar figures are proportional (have the same ratio).
Proportional Sides
- The ratio of corresponding sides in similar figures is constant.
- This ratio is known as the scale factor.
- If the scale factor is greater than 1, the larger figure is an enlargement of the smaller figure.
- If the scale factor is between 0 and 1, the smaller figure is a reduction of the larger figure.
Similar Polygons
- Two polygons are similar if their corresponding angles are congruent and their corresponding sides are proportional.
- The symbol for similarity is ~. For example, if polygon ABCDE is similar to polygon FGHIJ, we write ABCDE ~ FGHIJ.
- This applies to all polygons (triangles, quadrilaterals, pentagons, etc.).
Similar Triangles
- Similar triangles have the same shape, but potentially different sizes.
- A triangle can be proven similar to another if:
- Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
- Side-Side-Side (SSS) Similarity Theorem: If the corresponding sides of two triangles are proportional, then the triangles are similar.
- Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle, and the sides including those angles are proportional, then the triangles are similar.
Applications
- Similar figures and their properties are useful in many applications, including:
- Finding unknown lengths in similar figures.
- Calculating heights of objects (e.g., using shadows).
- Creating scale drawings.
- Designing models or blueprints.
- Solving geometric problems.
Key Concepts Summary
- Similarity involves figures with identical shapes but not necessarily equal sizes.
- Corresponding angles are equal, and corresponding sides are proportional.
- The scale factor relates the size of corresponding sides.
- Similarity is a fundamental concept in geometry with practical applications.
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