To solve for the length of the other diagonal [tex]\( \overline{D F} \)[/tex] in the kite, we need to follow the given steps:
1. Calculate the semi-perimeter [tex]\( s \)[/tex] of the kite:
One of the triangles formed by the kite can be considered with sides measuring 20 cm, 13 cm, and 24 cm. The semi-perimeter [tex]\( s \)[/tex] of this triangle is given by:
[tex]\[
s = \frac{a + b + c}{2}
\][/tex]
Here, [tex]\( a = 20 \)[/tex] cm, [tex]\( b = 13 \)[/tex] cm, and [tex]\( c = 24 \)[/tex] cm. So,
[tex]\[
s = \frac{20 + 13 + 24}{2} = 28.5 \text{ cm}
\][/tex]
2. Calculate the area of one of the triangles formed by the known diagonal using Heron's formula:
Heron's formula states that the area [tex]\( A \)[/tex] of a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is:
[tex]\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
Substituting the given values:
[tex]\[
A = \sqrt{28.5 \times (28.5 - 20) \times (28.5 - 13) \times (28.5 - 24)}
\][/tex]
Simplifying inside the square root:
[tex]\[
A = \sqrt{28.5 \times 8.5 \times 15.5 \times 4.5}
\][/tex]
Calculating this gives us:
[tex]\[
A \approx 129.988 \text{ square cm}
\][/tex]
3. Calculate the total area of the kite:
The kite is composed of two congruent triangles, each with an area of approximately 129.988 square cm. Therefore, the total area of the kite is:
[tex]\[
\text{Total Area} = 2 \times 129.988 \approx 259.976 \text{ square cm}
\][/tex]
4. Determine the length of the other diagonal [tex]\( \overline{D F} \)[/tex]:
The area of a kite can also be given by the formula:
[tex]\[
\text{Area} = \frac{1}{2} \times \overline{E G} \times \overline{D F}
\][/tex]
Rearranging to solve for [tex]\( \overline{D F} \)[/tex]:
[tex]\[
\overline{D F} = \frac{2 \times \text{Area}}{\overline{E G}}
\][/tex]
Substituting the known values:
[tex]\[
\overline{D F} = \frac{2 \times 259.976}{24} \approx 21.665 \text{ cm}
\][/tex]
Therefore, the length of the other diagonal [tex]\( \overline{D F} \)[/tex] is approximately [tex]\( 21.665 \)[/tex] cm. The closest answer choice is:
[tex]\[
\boxed{21 \text{ cm}}
\][/tex]
