How do you find the missing length of similar rectangles?

To find the missing length of similar rectangles, you primarily use the concept of a scale factor. Similar rectangles have the same shape but can differ in size; their corresponding sides are proportional.

Understanding Similar Rectangles and Scale Factor

Similar rectangles maintain proportional relationships between their corresponding sides. This means that if you divide the length of one side of the larger rectangle by the length of the corresponding side of the smaller rectangle, you will get the same ratio for all corresponding sides. This ratio is known as the scale factor.

What is a Scale Factor?

A scale factor is the constant multiplier that relates the dimensions of one similar figure to another. It tells you how much larger or smaller one shape is compared to the other.

• Calculating the Scale Factor: If you have two similar rectangles, you can find the scale factor by dividing the length of a side from the new (or larger) rectangle by the length of the corresponding side from the original (or smaller) rectangle.
  • Scale Factor = (Length of a side in Rectangle B) / (Corresponding length of a side in Rectangle A)

Steps to Find the Missing Length

Finding a missing length in similar rectangles is straightforward once you determine the scale factor.

1. Identify Corresponding Sides:
   • First, clearly identify which sides of the two similar rectangles correspond to each other. For rectangles, this usually means length to length and width to width.
2. Calculate the Scale Factor:
   • Use a pair of known corresponding sides from both rectangles to calculate the scale factor.
   • For example, if Rectangle A has a width of 3 units and Rectangle B has a corresponding width of 6 units, the scale factor would be 6 units / 3 units = 2.
3. Apply the Scale Factor:
   • Once you have the scale factor, multiply the known side of the original rectangle (the one corresponding to the missing side) by the scale factor to find the missing length.
   • For instance, if the scale factor is 2 and the corresponding known side on the smaller rectangle is 4 units, then the missing side on the larger rectangle would be 4 units × 2 = 8 units.

Practical Example

Let's say you have two similar rectangles: Rectangle P and Rectangle Q.

• Rectangle P: Length = 5 units, Width = 3 units
• Rectangle Q: Length = X (missing), Width = 9 units

Here's how to find the missing length (X):

1. Identify Corresponding Sides:

   • Width of P (3 units) corresponds to Width of Q (9 units).
   • Length of P (5 units) corresponds to Length of Q (X).

2. Calculate the Scale Factor:

   • Using the known corresponding widths:
     Scale Factor = Width of Q / Width of P
Scale Factor = 9 units / 3 units = 3

3. Apply the Scale Factor:

   • Now, multiply the length of Rectangle P by the scale factor to find the missing length X for Rectangle Q:
     Missing Length (X) = Length of P × Scale Factor
X = 5 units × 3
X = 15 units

Therefore, the missing length of Rectangle Q is 15 units.

This method allows you to efficiently determine any unknown side length in similar rectangles, simplifying geometric problems.

For more information on similar figures and proportions, you can explore resources on geometry and similar shapes.