To solve for the length of the other diagonal, [tex]\( \overline{DF} \)[/tex], in the kite, here's the step-by-step solution:
1. Analyze the Patterns of the Kite:
- We have a kite with one diagonal, [tex]\( \overline{EG} \)[/tex], measuring 24 cm.
- The top two sides of the kite each measure 20 cm, and the bottom two sides each measure 13 cm.
2. Properties of the Kite:
- In a kite, the diagonals intersect each other at right angles (90 degrees).
- The diagonals bisect each other.
3. Divide the Diagonal [tex]\( \overline{EG} \)[/tex]:
- Let's denote the length of [tex]\( \overline{EG} \)[/tex] as 24 cm.
- Since the diagonals bisect each other perpendicularly, half of [tex]\( \overline{EG} \)[/tex] is [tex]\( 24 \, \text{cm} / 2 = 12 \, \text{cm} \)[/tex].
- So each half of [tex]\( \overline{EG} \)[/tex] is 12 cm.
4. Using the Right Triangle Formed by the Diagonals:
- For the top part of the kite, each side measures 20 cm.
- Form a right triangle with one leg as half of [tex]\( \overline{EG} \)[/tex] (12 cm) and the hypotenuse as one of the top sides (20 cm).
5. Applying the Pythagorean Theorem:
- Let’s denote half of the other diagonal [tex]\( \overline{DF} \)[/tex] as [tex]\( x \)[/tex].
- Using the Pythagorean theorem in the triangle formed by the sides of the kite and the halves of the diagonals:
[tex]\[
(\text{side})^2 = (\text{one half of } \overline{DF})^2 + (\text{one half of } \overline{EG})^2
\][/tex]
- For the top side:
[tex]\[
20^2 = x^2 + 12^2
\][/tex]
- Simplifying:
[tex]\[
400 = x^2 + 144
\][/tex]
- Isolating [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = 400 - 144
\][/tex]
[tex]\[
x^2 = 256
\][/tex]
- Taking the square root of both sides:
[tex]\[
x = 16
\][/tex]
6. Full Length of [tex]\( \overline{ DF } \)[/tex]:
- Since [tex]\( x \)[/tex] is half of [tex]\( \overline{ DF } \)[/tex], the full length of [tex]\( \overline{ DF } \)[/tex] is:
[tex]\[
\overline{DF} = 2 \times 16 = 32 \, \text{cm}
\][/tex]
Therefore, the length of the other diagonal [tex]\( \overline{DF} \)[/tex] is 32 cm.
The correct answer is:
[tex]\[ \boxed{32 \, \text{cm}} \][/tex]
