Dilating an Isosceles Right Triangle

Oct 13, 2025 by miodeya.com 26 views

Introduction

In geometry, dilation is a transformation that changes the size of a figure. When dilating a figure, we need to determine the scale factor, which is the ratio of the new size to the original size. In this article, we will explore how to find the scale factor when dilating an isosceles right triangle.

What is an Isosceles Right Triangle?

An isosceles right triangle is a triangle with two equal sides and one right angle (90 degrees). The two equal sides are called legs, and the side opposite the right angle is called the hypotenuse. In an isosceles right triangle, the legs are equal in length, and the hypotenuse is equal to the square root of 2 times the length of a leg.

Dilating an Isosceles Right Triangle

When dilating an isosceles right triangle, we need to determine the scale factor, which is the ratio of the new size to the original size. To do this, we need to know the length of the hypotenuse of the original triangle and the length of the hypotenuse of the dilated triangle.

Given Information

In this problem, we are given that the hypotenuse of the original isosceles right triangle is 18 yards, and we want to dilate it to create a dilated image with a hypotenuse of 12 yards.

Finding the Scale Factor

To find the scale factor, we need to divide the length of the hypotenuse of the dilated triangle by the length of the hypotenuse of the original triangle.

Scale factor = (length of hypotenuse of dilated triangle) / (length of hypotenuse of original triangle)

Substituting the given values, we get:

Scale factor = 12 yards / 18 yards

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 6.

Scale factor = (12 yards / 6) / (18 yards / 6)

Scale factor = 2 / 3

Conclusion

In conclusion, to dilate an isosceles right triangle with a hypotenuse of 18 yards to create a dilated image with a hypotenuse of 12 yards, we need to use a scale factor of 2/3.

Answer

The correct answer is:

A. 2/3

Why is this the correct answer?

This is the correct answer because the scale factor is the ratio of the new size to the original size. In this case, the new size is 12 yards, and the original size is 18 yards. Therefore, the scale factor is 12 yards / 18 yards = 2/3.

What is the significance of this problem?

This problem is significant because it illustrates the concept of scale factor in geometry. Understanding scale factors is important in various real-world applications, such as architecture, engineering, and design.

Real-World Applications

Scale factors are used in various real-world applications, such as:

Architecture: Architects use scale factors to design buildings and structures that are proportional to the original design.
Engineering: Engineers use scale factors to design and build machines and systems that are proportional to the original design.
Design: Designers use scale factors to create designs that are proportional to the original design.

Conclusion

In conclusion, understanding scale factors is important in geometry and has various real-world applications. By using scale factors, we can create proportional designs and structures that are accurate and efficient.

Final Thoughts

In this article, we explored how to find the scale factor when dilating an isosceles right triangle. We also discussed the significance of scale factors in geometry and their real-world applications. By understanding scale factors, we can create accurate and efficient designs and structures that meet our needs.

References

[1] "Geometry" by Michael Artin
[2] "Mathematics for Computer Science" by Eric Lehman
[3] "Geometry: A Comprehensive Introduction" by Dan Pedoe

Additional Resources

[1] Khan Academy: Geometry
[2] Math Open Reference: Geometry
[3] Wolfram MathWorld: Geometry
Frequently Asked Questions (FAQs) =====================================

Q: What is a scale factor in geometry?

A: A scale factor is a ratio of the new size to the original size of a figure. It is used to describe the amount of enlargement or reduction of a figure.

Q: How do you find the scale factor of a dilated figure?

A: To find the scale factor of a dilated figure, you need to divide the length of the corresponding side of the dilated figure by the length of the corresponding side of the original figure.

Q: What is the difference between a scale factor and a ratio?

A: A scale factor is a ratio of the new size to the original size of a figure, while a ratio is a comparison of two numbers. A scale factor is a specific type of ratio that is used to describe the amount of enlargement or reduction of a figure.

Q: Can a scale factor be greater than 1?

A: Yes, a scale factor can be greater than 1. If the scale factor is greater than 1, it means that the figure has been enlarged. If the scale factor is less than 1, it means that the figure has been reduced.

Q: How do you determine if a figure has been enlarged or reduced?

A: To determine if a figure has been enlarged or reduced, you need to compare the scale factor to 1. If the scale factor is greater than 1, the figure has been enlarged. If the scale factor is less than 1, the figure has been reduced.

Q: What is the significance of scale factors in real-world applications?

A: Scale factors are used in various real-world applications, such as architecture, engineering, and design. They are used to create proportional designs and structures that are accurate and efficient.

Q: Can scale factors be used to dilate figures other than triangles?

A: Yes, scale factors can be used to dilate figures other than triangles. Any figure can be dilated using a scale factor, as long as the corresponding sides are known.

Q: How do you find the scale factor of a dilated figure with a hypotenuse of 15 yards and an original hypotenuse of 20 yards?

A: To find the scale factor of a dilated figure with a hypotenuse of 15 yards and an original hypotenuse of 20 yards, you need to divide the length of the hypotenuse of the dilated figure by the length of the hypotenuse of the original figure.

Scale factor = (length of hypotenuse of dilated figure) / (length of hypotenuse of original figure)

Substituting the given values, we get:

Scale factor = 15 yards / 20 yards

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 5.

Scale factor = (15 yards / 5) / (20 yards / 5)

Scale factor = 3 / 4

Q: What is the scale factor of a dilated figure with a hypotenuse of 24 yards and an original hypotenuse of 30 yards?

A: To find the scale factor of a dilated figure with a hypotenuse of 24 yards and an original hypotenuse of 30 yards, you need to divide the length of the hypotenuse of the dilated figure by the length of the hypotenuse of the original figure.

Scale factor = (length of hypotenuse of dilated figure) / (length of hypotenuse of original figure)

Substituting the given values, we get:

Scale factor = 24 yards / 30 yards

To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 6.

Scale factor = (24 yards / 6) / (30 yards / 6)

Scale factor = 4 / 5

Conclusion

In conclusion, scale factors are an important concept in geometry that is used to describe the amount of enlargement or reduction of a figure. By understanding scale factors, we can create proportional designs and structures that are accurate and efficient.