Ashley saved a distance equal to %80 of the length of the shortest side of a rectangular field by cutting across the diagonal of the field instead of along two of the sides. Find the ratio of the length of the shortest side of the field to the length of its longest side.

Let's break down the problem to find the ratio of the shortest side of the rectangular field to the longest side.

### Step 1: Label the dimensions
Let the shortest side of the rectangle be denoted as \( a \), and the longest side as \( b \). The diagonal of the rectangle is calculated using the Pythagorean theorem as:

\[
d = \sqrt{a^2 + b^2}
\]

### Step 2: Distance saved
Ashley saved 80% of the length of the shortest side by cutting across the diagonal. This means the distance along two sides is reduced by:

\[
\text{Distance saved} = 0.80 \times a
\]

### Step 3: Calculate the distance along two sides
The total distance along two sides is \( a + b \). So, the distance saved implies that the diagonal distance \( d \) is:

\[
d = (a + b) - 0.80 \times a
\]

Simplifying this:

\[
d = a + b - 0.80a = 0.20a + b
\]

### Step 4: Set up the equation
We now have two expressions for \( d \):
1. From the Pythagorean theorem: \( d = \sqrt{a^2 + b^2} \)
2. From the distance saved: \( d = 0.20a + b \)

Equating the two:

\[
\sqrt{a^2 + b^2} = 0.20a + b
\]

### Step 5: Square both sides
To eliminate the square root, square both sides of the equation:

\[
a^2 + b^2 = (0.20a + b)^2
\]

Expand the right-hand side:

\[
a^2 + b^2 = 0.04a^2 + 0.40ab + b^2
\]

### Step 6: Simplify the equation
Cancel \( b^2 \) from both sides:

\[
a^2 = 0.04a^2 + 0.40ab
\]

Move terms involving \( a^2 \) to the left:

\[
a^2 - 0.04a^2 = 0.40ab
\]

Simplify:

\[
0.96a^2 = 0.40ab
\]

### Step 7: Solve for \( \frac{a}{b} \)
Now divide both sides by \( 0.40ab \):

\[
\frac{a}{b} = \frac{0.40}{0.96}
\]

Simplify the fraction:

\[
\frac{a}{b} = \frac{5}{12}
\]

### Conclusion:
The ratio of the shortest side to the longest side of the rectangular field is:

\[
\boxed{\frac{5}{12}}
\]

Would you like more details or further clarification? Here are five related questions for practice:

1. If the shortest side of the rectangle is 10 units, what is the length of the longest side?
2. How does the Pythagorean theorem help in solving this problem?
3. What percentage of the distance is saved if the diagonal is used, but the sides have different proportions?
4. If the ratio of the sides is different, how would the equation change?
5. Can you generalize the problem for a square field?

**Tip:** Always draw a diagram when solving geometry problems involving distances. It helps visualize the relationships between sides, diagonals, and angles!

Ashley saved a distance equal to 80% of the length of the shortest side of a rectangular field by cutting across the diagonal of the field instead of along two of the sides. Find the ratio of the length of the shortest side of the field to the length of its longest side.

Learn how to find the ratio of the shortest side to the longest side of a rectangular field using the Pythagorean theorem. This problem explores key geometry concepts, including diagonal distances and distance savings. Suitable for students in Grades 9-12.

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