the length of one base of a trapezoid is four times the length of the other base. If the length of the median is 40, find the length of each base.

To solve this problem, we will use the property of a trapezoid where the length of the median (the line segment connecting the midpoints of the non-parallel sides) is the average of the lengths of the two bases.

### Step 1: Define the variables
Let:
- \( b_1 \) be the length of the shorter base.
- \( b_2 \) be the length of the longer base.

According to the problem, the longer base is four times the length of the shorter base. So we can write:
\[
b_2 = 4b_1
\]

### Step 2: Use the median formula
The length of the median of a trapezoid is the average of the lengths of the two bases:
\[
\text{Median} = \frac{b_1 + b_2}{2}
\]
We are given that the median is 40, so we can set up the equation:
\[
40 = \frac{b_1 + b_2}{2}
\]

### Step 3: Substitute \( b_2 = 4b_1 \) into the equation
Substitute \( b_2 \) with \( 4b_1 \) into the median equation:
\[
40 = \frac{b_1 + 4b_1}{2}
\]
Simplifying:
\[
40 = \frac{5b_1}{2}
\]

### Step 4: Solve for \( b_1 \)
Multiply both sides of the equation by 2 to eliminate the fraction:
\[
80 = 5b_1
\]
Now, divide both sides by 5:
\[
b_1 = 16
\]

### Step 5: Find \( b_2 \)
Since \( b_2 = 4b_1 \), we have:
\[
b_2 = 4 \times 16 = 64
\]

### Final Answer:
The lengths of the bases are:
- \( b_1 = 16 \) (shorter base)
- \( b_2 = 64 \) (longer base)

Let me know if you'd like further clarification on any step or if you have any other questions! 

### Related questions:
1. How does the area of a trapezoid relate to its bases and height?
2. What is the formula for the area of a trapezoid?
3. How would the solution change if the trapezoid was isosceles?
4. Can you calculate the height of the trapezoid if you know the area?
5. What other geometric properties can we find for trapezoids based on base lengths?

**Tip:** When dealing with trapezoids, always remember that the median is the average of the two bases—this is a helpful shortcut for many problems!

The length of one base of a trapezoid is four times the length of the other base. If the length of the median is 40, find the length of each base.

Learn how to find the lengths of the bases of a trapezoid when the median is given. This step-by-step solution involves using the properties of trapezoids and algebraic equations to solve for the lengths of the bases. Suitable for students in Grades 7-9.

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