Quiz 6 1 Ratios And Similar Figures Answer Key

Quiz 6.1: Ratios and Similar Figures - Answer Key & Comprehensive Guide

This comprehensive guide provides detailed answers and explanations for Quiz 6.1 on ratios and similar figures, a topic fundamental to geometry and crucial for various applications in mathematics and beyond. We’ll delve into the core concepts, provide solutions to sample problems, and offer strategies for mastering this subject. This guide aims to not only provide the answer key but also to enhance your understanding of ratios and similar figures.

Understanding Ratios

Before diving into the quiz, let's solidify our understanding of ratios. A ratio is a comparison of two or more quantities. It shows the relative size of one quantity to another. Ratios can be expressed in several ways:

• Using a colon: a:b (read as "a to b")
• As a fraction: a/b
• Using the word "to": a to b

Example: If there are 3 red balls and 5 blue balls, the ratio of red balls to blue balls is 3:5, 3/5, or 3 to 5.

Important Note: Ratios are usually expressed in their simplest form. This means reducing the fraction to its lowest terms. For example, the ratio 6:12 simplifies to 1:2.

Similar Figures: The Foundation

Similar figures are figures that have the same shape but different sizes. This means that their corresponding angles are congruent (equal), and their corresponding sides are proportional. This proportionality is where ratios play a crucial role.

The ratio of corresponding sides of similar figures is called the scale factor. This scale factor is constant for all corresponding sides.

Key Properties of Similar Figures

• Congruent Angles: Corresponding angles in similar figures are equal.
• Proportional Sides: Corresponding sides in similar figures are proportional. This means the ratio of the lengths of corresponding sides remains constant.
• Scale Factor: The ratio of corresponding side lengths of similar figures is the scale factor.

Solving Problems Involving Ratios and Similar Figures

Let's tackle some typical problems encountered in Quiz 6.1, illustrating the application of ratios and the properties of similar figures.

Problem 1: Finding the Scale Factor

Two similar triangles, ΔABC and ΔDEF, have side lengths AB = 6 cm, BC = 8 cm, AC = 10 cm, and DE = 3 cm, EF = 4 cm, DF = 5 cm. Find the scale factor.

Solution:

Compare the corresponding sides:

• AB/DE = 6/3 = 2
• BC/EF = 8/4 = 2
• AC/DF = 10/5 = 2

The scale factor is 2. ΔABC is twice as large as ΔDEF.

Problem 2: Finding Missing Side Lengths

Two similar rectangles, ABCD and EFGH, have lengths AB = 12 cm and EF = 6 cm. If the width of rectangle ABCD is 8 cm, what is the width of rectangle EFGH?

Solution:

First, find the scale factor: AB/EF = 12/6 = 2

Since the rectangles are similar, the ratio of their widths is also 2. Let x be the width of rectangle EFGH.

8/x = 2

x = 8/2 = 4 cm

The width of rectangle EFGH is 4 cm.

Problem 3: Area and Similar Figures

Two similar squares have side lengths of 5 cm and 10 cm. What is the ratio of their areas?

Solution:

The area of the first square is 5² = 25 cm². The area of the second square is 10² = 100 cm².

The ratio of their areas is 25/100 = 1/4.

Important Note: The ratio of the areas of similar figures is the square of the scale factor. In this case, the scale factor is 10/5 = 2, and the ratio of the areas is 2² = 4.

Problem 4: Volume and Similar Figures

Two similar cubes have side lengths of 3 cm and 6 cm. What is the ratio of their volumes?

Solution:

The volume of the first cube is 3³ = 27 cm³. The volume of the second cube is 6³ = 216 cm³.

The ratio of their volumes is 27/216 = 1/8.

Important Note: The ratio of the volumes of similar figures is the cube of the scale factor. In this case, the scale factor is 6/3 = 2, and the ratio of the volumes is 2³ = 8.

Problem 5: Word Problem Application

A tree casts a shadow 15 meters long. At the same time, a 2-meter tall person casts a shadow 3 meters long. How tall is the tree?

Solution:

We can set up a proportion using similar triangles formed by the tree, its shadow, and the person and their shadow:

Height of tree / Length of tree's shadow = Height of person / Length of person's shadow

Let h be the height of the tree:

h / 15 = 2 / 3

h = (2 * 15) / 3 = 10 meters

The tree is 10 meters tall.

Quiz 6.1: Sample Questions and Answers (with detailed explanations)

This section provides sample questions mirroring the style and difficulty of Quiz 6.1, along with thorough explanations of the solutions. Remember to always show your work, even on seemingly simple problems – this demonstrates your understanding and helps identify potential mistakes.

Question 1: What is the ratio of 15 to 25 in simplest form?

Answer: 3/5 or 3:5. Both 15 and 25 are divisible by 5.

Question 2: Two similar triangles have a scale factor of 3:2. If one triangle has a side length of 9 cm, what is the corresponding side length in the other triangle?

Answer: Let x be the corresponding side length. The ratio is 3/2 = 9/x. Solving for x, we get x = 6 cm.

Question 3: The ratio of boys to girls in a class is 2:3. If there are 12 boys, how many girls are there?

Answer: 2/3 = 12/x. Solving for x, we get x = 18 girls.

Question 4: Two similar rectangles have areas of 36 cm² and 16 cm². What is the ratio of their perimeters?

Answer: The ratio of the areas is 36/16 = 9/4. Since the ratio of the areas is the square of the scale factor, the scale factor is √(9/4) = 3/2. The ratio of their perimeters is equal to the scale factor, which is 3:2.

Question 5: Two similar cubes have a volume ratio of 8:27. What is the ratio of their corresponding side lengths?

Answer: The volume ratio is the cube of the scale factor. Therefore, the scale factor is the cube root of 8/27, which is 2/3.

Question 6: A map has a scale of 1 cm : 10 km. If the distance between two cities on the map is 5 cm, what is the actual distance between the cities?

Answer: The scale is 1 cm / 10 km = 5 cm / x km. Solving for x, we get x = 50 km.

Question 7: Two similar pentagons have perimeters of 20 cm and 30 cm. What is the ratio of their areas?

Answer: The ratio of their perimeters is 20/30 = 2/3. This is also the scale factor. The ratio of their areas is the square of the scale factor, which is (2/3)² = 4/9.

Question 8: A 6-foot tall man casts a shadow that is 8 feet long. At the same time, a building casts a shadow of 40 feet. How tall is the building?

Answer: Set up a proportion: 6/8 = x/40. Solving for x, we get x = 30 feet.

Mastering Ratios and Similar Figures: Tips and Strategies

• Practice Regularly: Consistent practice is key to mastering any mathematical concept. Solve a variety of problems, including word problems, to build your skills.
• Understand the Concepts: Don't just memorize formulas; make sure you grasp the underlying principles of ratios and similar figures.
• Draw Diagrams: Visualizing problems through diagrams can greatly aid your understanding and problem-solving process, especially with geometry problems.
• Check Your Work: Always review your calculations and ensure your answers make sense in the context of the problem.
• Use Online Resources: Explore online resources like educational websites and videos to reinforce your learning.
• Seek Help When Needed: Don't hesitate to ask your teacher or tutor for assistance if you're struggling with a particular concept.

By diligently applying these strategies and consistently reviewing the concepts outlined in this guide, you'll build a solid understanding of ratios and similar figures and confidently tackle any quiz or exam related to these topics. Remember that consistent effort and a clear understanding of the fundamental principles are the keys to success in mathematics.