To find the length of the other diagonal [tex]\(\overline{DF}\)[/tex] of the kite, we need to work through the information given about the kite step by step.
1. Identify the given dimensions:
- Top sides of the kite: 20 cm each
- Bottom sides of the kite: 13 cm each
- One diagonal [tex]\(\overline{EG}\)[/tex]: 24 cm
2. Understand the properties of the kite:
- The diagonals of the kite intersect at right angles (90 degrees) and bisect each other.
3. Divide the given diagonal [tex]\(\overline{EG}\)[/tex] into two equal parts:
[tex]\[
\overline{EG} = 24 \text{ cm}
\][/tex]
Half of [tex]\(\overline{EG}\)[/tex] would be:
[tex]\[
x = \frac{\overline{EG}}{2} = \frac{24}{2} = 12 \text{ cm}
\][/tex]
4. Use the Pythagorean Theorem for the right triangles formed:
- Let's consider one of the right triangles formed by the sides of the kite and half of the diagonal [tex]\(\overline{EG}\)[/tex]:
- For the right triangle with hypotenuse (side of the kite) 20 cm:
[tex]\[
y^2 + x^2 = 20^2
\][/tex]
[tex]\[
y^2 + 12^2 = 400
\][/tex]
[tex]\[
y^2 + 144 = 400
\][/tex]
[tex]\[
y^2 = 400 - 144 = 256
\][/tex]
[tex]\[
y = \sqrt{256} = 16 \text{ cm}
\][/tex]
5. Determine one half of the other diagonal:
[tex]\[
y = 16 \text{ cm}
\][/tex]
Since diagonals bisect each other at right angles, and we need the full length of [tex]\(\overline{DF}\)[/tex]:
[tex]\[
\overline{DF} = 2 \times y = 2 \times 16 = 32 \text{ cm}
\][/tex]
Therefore, the length of the other diagonal [tex]\(\overline{DF}\)[/tex] is:
[tex]\[
\boxed{32 \text{ cm}}
\][/tex]
