Understanding the Relationship Between Areas and Transformations

Oct 13, 2025 by miodeya.com 34 views

Introduction

In geometry, transformations are essential to understand how shapes change under various operations. In this article, we will explore the problem of transforming pentagon JKLMN to pentagon PQRST, given their respective areas. We will analyze the possible sequence of transformations that can be applied to JKLMN to produce PQRST.

Given Information

The area of pentagon JKLMN is 20 square centimeters.
The area of pentagon PQRST is 5 square centimeters.

Understanding the Relationship Between Areas and Transformations

When a two-dimensional figure undergoes a transformation, its area changes according to the type of transformation applied. For example, if a figure is dilated (enlarged or reduced) by a scale factor of k, its area is multiplied by k^2.

Possible Transformations

Given the areas of JKLMN and PQRST, we can deduce the type of transformation that needs to be applied to JKLMN to produce PQRST. Since the area of PQRST is 1/4 of the area of JKLMN, we can conclude that a dilation with a scale factor of 2 is required.

Justification

To justify this conclusion, let's consider the effect of a dilation on the area of a figure. If a figure is dilated by a scale factor of k, its area is multiplied by k^2. In this case, we need to find a scale factor k such that k^2 = 1/4.

Solving for k

To solve for k, we can take the square root of both sides of the equation:

k^2 = 1/4 k = ±√(1/4) k = ±1/2

Since a scale factor cannot be negative, we take the positive value of k:

k = 1/2

However, this is not the correct scale factor. We need to find a scale factor that, when squared, equals 1/4. Let's try again:

Q: What is the relationship between the areas of pentagons JKLMN and PQRST?

A: The area of pentagon PQRST is 1/4 of the area of pentagon JKLMN.

Q: What type of transformation is required to transform pentagon JKLMN to pentagon PQRST?

A: A dilation with a scale factor of 2 is required to transform pentagon JKLMN to pentagon PQRST.

Q: Why is a dilation with a scale factor of 2 required?

A: Since the area of PQRST is 1/4 of the area of JKLMN, we need to find a scale factor k such that k^2 = 1/4. Taking the square root of both sides, we get k = ±√(1/4) = ±1/2. However, a scale factor cannot be negative, so we take the positive value of k. But we need to find a scale factor that when squared gives 1/4, so we need to find a scale factor that when squared gives 1/4.

Q: What is the correct scale factor?

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