Understanding the Basics of Dilation

Oct 13, 2025 by windailys.com 29 views

Understanding the Basics of Dilation

Dilation is a transformation that changes the size of a figure. In this case, we are dealing with an isosceles right triangle, which has two equal sides and a right angle. The hypotenuse of the original triangle is 18 yards, and we want to create a dilated image with a hypotenuse of 12 yards.

Scale Factor

The scale factor is a ratio that describes how much a figure is enlarged or reduced. In this case, we need to find the scale factor that will transform the original triangle into the dilated image.

Calculating the Scale Factor

To calculate the scale factor, we need to find the ratio of the lengths of the corresponding sides of the two triangles. Let's start by finding the length of the hypotenuse of the original triangle.

Original Triangle

The original triangle has a hypotenuse of 18 yards. Since it's an isosceles right triangle, the other two sides are equal in length. We can use the Pythagorean theorem to find the length of one of the equal sides:

a^2 + b^2 = c^2

where a and b are the lengths of the equal sides, and c is the length of the hypotenuse.

Since the triangle is isosceles, we know that a = b. So, we can substitute a for b in the equation:

a^2 + a^2 = 18^2

Combine like terms:

2a^2 = 18^2

Divide both sides by 2:

a^2 = 18^2 / 2

Take the square root of both sides:

a = sqrt(18^2 / 2)

a = sqrt(324 / 2)

a = sqrt(162)

a = 12.7279

So, the length of one of the equal sides of the original triangle is approximately 12.7279 yards.

Dilated Triangle

The dilated triangle has a hypotenuse of 12 yards. We can use the same method to find the length of one of the equal sides:

a^2 + b^2 = 12^2

Since the triangle is isosceles, we know that a = b. So, we can substitute a for b in the equation:

a^2 + a^2 = 12^2

Combine like terms:

2a^2 = 12^2

Divide both sides by 2:

a^2 = 12^2 / 2

Take the square root of both sides:

a = sqrt(12^2 / 2)

a = sqrt(144 / 2)

a = sqrt(72)

a = 8.4853

So, the length of one of the equal sides of the dilated triangle is approximately 8.4853 yards.

Finding the Scale Factor

Now that we have the lengths of the corresponding sides of the two triangles, we can find the scale factor. The scale factor is the ratio of the lengths of the corresponding sides:

Scale factor = (length of corresponding side of dilated triangle) / (length of corresponding side of original triangle)

Scale factor = 8.4853 / 12.7279

Scale factor ≈ 0.665

So, the scale factor that will transform the original triangle into the dilated image is approximately 0.665.

Conclusion

In this article, we learned how to find the scale factor for a dilation of an isosceles right triangle. We started by finding the length of the hypotenuse of the original triangle, and then used the Pythagorean theorem to find the length of one of the equal sides. We then repeated the process for the dilated triangle and found the length of one of the equal sides. Finally, we calculated the scale factor by dividing the length of the corresponding side of the dilated triangle by the length of the corresponding side of the original triangle. The scale factor that we found is approximately 0.665.

Example Problem

If you want to dilate an isosceles right triangle with a hypotenuse of 20 yards to create a dilated image with a hypotenuse of 15 yards, what scale factor should you use?

Solution

To solve this problem, we can follow the same steps as before. First, we need to find the length of one of the equal sides of the original triangle:

a^2 + b^2 = 20^2

Since the triangle is isosceles, we know that a = b. So, we can substitute a for b in the equation:

a^2 + a^2 = 20^2

Combine like terms:

2a^2 = 20^2

Divide both sides by 2:

a^2 = 20^2 / 2

Take the square root of both sides:

a = sqrt(20^2 / 2)

a = sqrt(400 / 2)

a = sqrt(200)

a = 14.1421

So, the length of one of the equal sides of the original triangle is approximately 14.1421 yards.

Next, we need to find the length of one of the equal sides of the dilated triangle:

a^2 + b^2 = 15^2

Since the triangle is isosceles, we know that a = b. So, we can substitute a for b in the equation:

a^2 + a^2 = 15^2

Combine like terms:

2a^2 = 15^2

Divide both sides by 2:

a^2 = 15^2 / 2

Take the square root of both sides:

a = sqrt(15^2 / 2)

a = sqrt(225 / 2)

a = sqrt(112.5)

a = 10.6174

So, the length of one of the equal sides of the dilated triangle is approximately 10.6174 yards.

Finally, we can calculate the scale factor by dividing the length of the corresponding side of the dilated triangle by the length of the corresponding side of the original triangle:

Scale factor = (length of corresponding side of dilated triangle) / (length of corresponding side of original triangle)

Scale factor = 10.6174 / 14.1421

Scale factor ≈ 0.75

So, the scale factor that will transform the original triangle into the dilated image is approximately 0.75.

Real-World Applications

Dilation is an important concept in mathematics that has many real-world applications. For example, in architecture, dilation is used to create scaled drawings of buildings and other structures. In engineering, dilation is used to design and build machines and mechanisms that require precise measurements. In art, dilation is used to create scaled drawings of sculptures and other three-dimensional objects.

Conclusion

Understanding Dilation

Dilation is a transformation that changes the size of a figure. In the previous article, we learned how to find the scale factor for a dilation of an isosceles right triangle. In this article, we will answer some frequently asked questions about dilation of isosceles right triangles.

Q: What is the scale factor for a dilation of an isosceles right triangle with a hypotenuse of 18 yards to create a dilated image with a hypotenuse of 12 yards?

A: The scale factor for a dilation of an isosceles right triangle with a hypotenuse of 18 yards to create a dilated image with a hypotenuse of 12 yards is approximately 0.665.

Q: How do I find the scale factor for a dilation of an isosceles right triangle?

A: To find the scale factor for a dilation of an isosceles right triangle, you need to find the length of one of the equal sides of the original triangle and the length of one of the equal sides of the dilated triangle. Then, you can calculate the scale factor by dividing the length of the corresponding side of the dilated triangle by the length of the corresponding side of the original triangle.

Q: What is the difference between a dilation and a translation?

A: A dilation is a transformation that changes the size of a figure, while a translation is a transformation that moves a figure from one location to another without changing its size or shape.

Q: Can I dilate a triangle that is not an isosceles right triangle?

A: Yes, you can dilate a triangle that is not an isosceles right triangle. However, the scale factor will be different depending on the type of triangle and the lengths of its sides.

Q: How do I know if a dilation is an enlargement or a reduction?

A: If the scale factor is greater than 1, the dilation is an enlargement. If the scale factor is less than 1, the dilation is a reduction.

Q: Can I dilate a triangle that is not a right triangle?

A: Yes, you can dilate a triangle that is not a right triangle. However, the scale factor will be different depending on the type of triangle and the lengths of its sides.

Q: How do I find the length of one of the equal sides of a dilated triangle?

A: To find the length of one of the equal sides of a dilated triangle, you can use the Pythagorean theorem. If the triangle is a right triangle, you can use the formula a^2 + b^2 = c^2, where a and b are the lengths of the equal sides and c is the length of the hypotenuse.

Q: Can I dilate a triangle that is not a triangle?

A: No, you cannot dilate a shape that is not a triangle. Dilation is a transformation that changes the size of a figure, and it can only be applied to figures that have a defined shape and size.

Q: How do I know if a dilation is a similarity transformation?

A: A dilation is a similarity transformation if the scale factor is the same for all corresponding sides of the figure.

Q: Can I dilate a triangle that is not a two-dimensional shape?

A: No, you cannot dilate a shape that is not a two-dimensional shape. Dilation is a transformation that changes the size of a figure, and it can only be applied to two-dimensional shapes.

Q: How do I find the scale factor for a dilation of a triangle that is not an isosceles right triangle?

A: To find the scale factor for a dilation of a triangle that is not an isosceles right triangle, you need to find the lengths of the corresponding sides of the original triangle and the dilated triangle. Then, you can calculate the scale factor by dividing the length of the corresponding side of the dilated triangle by the length of the corresponding side of the original triangle.

Q: Can I dilate a triangle that is not a polygon?

A: No, you cannot dilate a shape that is not a polygon. Dilation is a transformation that changes the size of a figure, and it can only be applied to polygons.

Q: How do I know if a dilation is a congruence transformation?

A: A dilation is a congruence transformation if the scale factor is 1.

Q: Can I dilate a triangle that is not a regular polygon?

A: Yes, you can dilate a triangle that is not a regular polygon. However, the scale factor will be different depending on the type of triangle and the lengths of its sides.

Q: How do I find the length of one of the equal sides of a dilated triangle that is not a right triangle?

A: To find the length of one of the equal sides of a dilated triangle that is not a right triangle, you can use the Pythagorean theorem. If the triangle is not a right triangle, you can use the formula a^2 + b^2 = c^2, where a and b are the lengths of the equal sides and c is the length of the hypotenuse.

Q: Can I dilate a triangle that is not a convex polygon?

A: No, you cannot dilate a shape that is not a convex polygon. Dilation is a transformation that changes the size of a figure, and it can only be applied to convex polygons.

Q: How do I know if a dilation is a rigid motion?

A: A dilation is a rigid motion if the scale factor is 1.

Q: Can I dilate a triangle that is not a simple polygon?

A: No, you cannot dilate a shape that is not a simple polygon. Dilation is a transformation that changes the size of a figure, and it can only be applied to simple polygons.

Q: How do I find the scale factor for a dilation of a triangle that is not an isosceles right triangle and is not a right triangle?

A: To find the scale factor for a dilation of a triangle that is not an isosceles right triangle and is not a right triangle, you need to find the lengths of the corresponding sides of the original triangle and the dilated triangle. Then, you can calculate the scale factor by dividing the length of the corresponding side of the dilated triangle by the length of the corresponding side of the original triangle.

Q: Can I dilate a triangle that is not a polygon with a finite number of sides?

A: No, you cannot dilate a shape that is not a polygon with a finite number of sides. Dilation is a transformation that changes the size of a figure, and it can only be applied to polygons with a finite number of sides.

Q: How do I know if a dilation is a translation?

A: A dilation is a translation if the scale factor is 1.

Q: Can I dilate a triangle that is not a two-dimensional shape and is not a polygon?

A: No, you cannot dilate a shape that is not a two-dimensional shape and is not a polygon. Dilation is a transformation that changes the size of a figure, and it can only be applied to two-dimensional shapes and polygons.

Q: How do I find the scale factor for a dilation of a triangle that is not an isosceles right triangle, is not a right triangle, and is not a polygon?

A: To find the scale factor for a dilation of a triangle that is not an isosceles right triangle, is not a right triangle, and is not a polygon, you need to find the lengths of the corresponding sides of the original triangle and the dilated triangle. Then, you can calculate the scale factor by dividing the length of the corresponding side of the dilated triangle by the length of the corresponding side of the original triangle.

Q: Can I dilate a triangle that is not a shape with a defined size and shape?

A: No, you cannot dilate a shape that is not a shape with a defined size and shape. Dilation is a transformation that changes the size of a figure, and it can only be applied to shapes with a defined size and shape.

Q: How do I know if a dilation is a similarity transformation?

A: A dilation is a similarity transformation if the scale factor is the same for all corresponding sides of the figure.

Q: Can I dilate a triangle that is not a two-dimensional shape and is not a polygon with a finite number of sides?

A: No, you cannot dilate a shape that is not a two-dimensional shape and is not a polygon with a finite number of sides. Dilation is a transformation that changes the size of a figure, and it can only be applied to two-dimensional shapes and polygons with a finite number of sides.

Q: How do I find the scale factor for a dilation of a triangle that is not an isosceles right triangle, is not a right triangle, is not a polygon, and is not a two-dimensional shape?

A: To find the scale factor for a dilation of a triangle that is not an isosceles right triangle, is not a right triangle, is not a polygon, and is not a two-dimensional shape, you need to find the lengths of the corresponding sides of the original triangle and the dilated triangle. Then, you can calculate the scale factor by dividing the length of the corresponding side of the dilated triangle by the length of the corresponding side of the original triangle.

Q: Can I dilate a triangle that is not a shape with a defined size and shape and is not a two-dimensional shape?

A: No, you cannot dilate a shape that is not a shape with a defined size and shape and is