Calculating the scale factor when enlarging a shape is an important skill in geometry. Once you understand it, it’s actually pretty simple! I remember the first time I learned this in Year 8; it felt like everything clicked. Let’s go through it step by step.
First, let’s talk about what enlargement means.
An enlargement is a change that makes a shape bigger (or sometimes smaller) but keeps the same proportions.
When you enlarge a shape, every point moves away from a specific spot called the centre of enlargement.
The scale factor is a number that tells you how much larger or smaller a shape will be after being enlarged.
Here’s how to find the scale factor when you enlarge a shape:
Look at the Original Shape and the Enlarged Shape: First, you need both shapes: the original one and the new, bigger one. Imagine you have a triangle, and now it looks bigger after the enlargement.
Choose a Point: Pick a specific point on the original shape—usually, a corner (or vertex) works best.
Find the New Point: See where that point has moved to in the enlarged shape.
Measure the Distances: You need to find the distance from the centre of enlargement to both the original point and its new point.
[ \text{Distance to original point} = \sqrt{(x_1 - x_c)^2 + (y_1 - y_c)^2} ]
[ \text{Distance to enlarged point} = \sqrt{(x_2 - x_c)^2 + (y_2 - y_c)^2} ]
Here, (xₐ, yₐ) is the location of the centre of enlargement.
Calculate the Scale Factor: Now, divide the distance to the enlarged point by the distance to the original point:
[ \text{Scale Factor} = \frac{\text{Distance to enlarged point}}{\text{Distance to original point}} ]
Let’s say your original triangle has a corner at (2, 3), the new triangle's corner is at (6, 9), and the centre of enlargement is at (0, 0).
For the original point (2, 3):
[ \text{Distance} = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13} ]
For the enlarged point (6, 9):
[ \text{Distance} = \sqrt{(6 - 0)^2 + (9 - 0)^2} = \sqrt{36 + 81} = \sqrt{117} ]
Now, we can find the scale factor:
[ \text{Scale Factor} = \frac{\sqrt{117}}{\sqrt{13}} = \sqrt{9} = 3 ]
So, the shape was enlarged by a scale factor of 3!
Remember to practice with different shapes and sizes. The more you do this, the easier it will be! Try drawing some shapes and calculating the scale factors. It’s a fun way to see how enlargements work!
Calculating the scale factor when enlarging a shape is an important skill in geometry. Once you understand it, it’s actually pretty simple! I remember the first time I learned this in Year 8; it felt like everything clicked. Let’s go through it step by step.
First, let’s talk about what enlargement means.
An enlargement is a change that makes a shape bigger (or sometimes smaller) but keeps the same proportions.
When you enlarge a shape, every point moves away from a specific spot called the centre of enlargement.
The scale factor is a number that tells you how much larger or smaller a shape will be after being enlarged.
Here’s how to find the scale factor when you enlarge a shape:
Look at the Original Shape and the Enlarged Shape: First, you need both shapes: the original one and the new, bigger one. Imagine you have a triangle, and now it looks bigger after the enlargement.
Choose a Point: Pick a specific point on the original shape—usually, a corner (or vertex) works best.
Find the New Point: See where that point has moved to in the enlarged shape.
Measure the Distances: You need to find the distance from the centre of enlargement to both the original point and its new point.
[ \text{Distance to original point} = \sqrt{(x_1 - x_c)^2 + (y_1 - y_c)^2} ]
[ \text{Distance to enlarged point} = \sqrt{(x_2 - x_c)^2 + (y_2 - y_c)^2} ]
Here, (xₐ, yₐ) is the location of the centre of enlargement.
Calculate the Scale Factor: Now, divide the distance to the enlarged point by the distance to the original point:
[ \text{Scale Factor} = \frac{\text{Distance to enlarged point}}{\text{Distance to original point}} ]
Let’s say your original triangle has a corner at (2, 3), the new triangle's corner is at (6, 9), and the centre of enlargement is at (0, 0).
For the original point (2, 3):
[ \text{Distance} = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13} ]
For the enlarged point (6, 9):
[ \text{Distance} = \sqrt{(6 - 0)^2 + (9 - 0)^2} = \sqrt{36 + 81} = \sqrt{117} ]
Now, we can find the scale factor:
[ \text{Scale Factor} = \frac{\sqrt{117}}{\sqrt{13}} = \sqrt{9} = 3 ]
So, the shape was enlarged by a scale factor of 3!
Remember to practice with different shapes and sizes. The more you do this, the easier it will be! Try drawing some shapes and calculating the scale factors. It’s a fun way to see how enlargements work!