To find the length of the other diagonal, [tex]\(\overline{DF}\)[/tex], we can use the properties of kites and the Pythagorean theorem. Here's the detailed step-by-step solution:
### Step 1: Understand the Kite Properties
A kite has two pairs of adjacent sides that are equal in length. In this problem, the top sides each measure 20 cm and the bottom sides each measure 13 cm.
### Step 2: Divide the Kite in Half
The diagonals of a kite are perpendicular and bisect each other. Thus, diagonal [tex]\(\overline{DF}\)[/tex] and diagonal [tex]\(\overline{EG}\)[/tex] intersect at a right angle and split the kite into four right-angled triangles.
### Step 3: Work with Diagonal [tex]\(\overline{EG}\)[/tex]
Diagonal [tex]\(\overline{EG}\)[/tex] measures 24 cm. When bisected by [tex]\(\overline{DF}\)[/tex], it creates two segments of equal length:
[tex]\[ \overline{EG}/2 = \frac{24}{2} = 12 \, \text{cm} \][/tex]
### Step 4: Identify the Lengths in the Right Triangles
Consider the right triangle formed by the half-diagonal segment (12 cm), half of [tex]\(\overline{DF}\)[/tex] (call it [tex]\(x\)[/tex]), and one of the sides of the kite.
1. For the top part of the kite:
- The hypotenuse (one of the top sides) is 20 cm.
- One leg is the segment of [tex]\(\overline{EG}\)[/tex], which is 12 cm.
- The other leg is [tex]\(\overline{DF}/2 = x\)[/tex].
2. For the bottom part of the kite:
- The hypotenuse (one of the bottom sides) is 13 cm.
- One leg is the segment of [tex]\(\overline{EG}\)[/tex], which is 12 cm.
- The other leg is [tex]\(\overline{DF}/2 = x\)[/tex].
### Step 5: Apply the Pythagorean Theorem
For the top triangle with side 20 cm:
[tex]\[
20^2 = 12^2 + x^2
\][/tex]
[tex]\[
400 = 144 + x^2 \\
x^2 = 400 - 144 \\
x^2 = 256 \\
x = \sqrt{256} \\
x = 16 \, \text{cm}
\][/tex]
For the bottom triangle with side 13 cm:
[tex]\[
13^2 = 12^2 + x^2
\][/tex]
[tex]\[
169 = 144 + x^2 \\
x^2 = 169 - 144 \\
x^2 = 25 \\
x = \sqrt{25} \\
x = 5 \, \text{cm}
\][/tex]
### Step 6: Solve for the Total Length of [tex]\(\overline{DF}\)[/tex]
Since [tex]\(\overline{DF}\)[/tex] is composed of two segments, each denoted as [tex]\(x\)[/tex], we need to find the appropriate [tex]\(x\)[/tex] value:
The larger [tex]\(x = 16 \, \text{cm}\)[/tex]. The other part of [tex]\(\overline{DF}\)[/tex] corresponds to the sum of both [tex]\(x\)[/tex] values:
[tex]\[
\text{Length of } \overline{DF} = 16 \, \text{cm} \text{ (top part)} + 16 \, \text{cm} \text{ (bottom part)} = 20\sqrt{3} \, \text{cm}
\][/tex]
Hence, the length of the other diagonal, [tex]\(\overline{DF}\)[/tex], is not a usual integer length but can be exactly calculated as [tex]\(20\sqrt{3}\)[/tex].
### Conclusion
From the given options, the closest fit that meets our requirements in a straightforward educational problem though more likely represents [tex]\(\boxed{32 \, \text{cm}}\)[/tex]` especially considering the actual correct values produce 20sqrt(3), calculated similarly no integer represents the close return.
Optionally observe that [tex]\(16 \, \text{cm}\)[/tex] close individual segment consideration but if fewer options work too שת^missing,
Hence:
The length of the other diagonal, [tex]\(\overline{DF}\)[/tex], is actually close to [tex]\(\boxed{32} \, \text{cm}\)[/tex]. if rounded nearest segment appoximations holding 20side measurements as reflection similarly other consideration\(20\sqrt(n)\Matching intermediate steps:
