To find the length of the other diagonal, [tex]\(\overline{DF}\)[/tex], we need to apply the Pythagorean theorem. Let's break the problem down into clear steps.
1. Identify Given Information:
- The kite has symmetrical properties.
- The top two sides of the kite each measure 20 cm.
- The bottom two sides of the kite each measure 13 cm.
- One diagonal, [tex]\(\overline{E G}\)[/tex], measures 24 cm.
2. Analyze the Kite Structure:
- Let’s consider the point where the diagonals intersect as the center, breaking each diagonal into two equal parts.
- Diagonal [tex]\(\overline{EG}\)[/tex] is given to be 24 cm, so each half of this diagonal is [tex]\( \frac{24}{2} = 12 \)[/tex] cm.
3. Form Two Right Triangles from the Top Part:
- For the right triangle from the top part with the hypotenuse of 20 cm (one side of the kite):
- One leg is half of [tex]\(\overline{EG}\)[/tex] = 12 cm.
- Let [tex]\(b_{\text{top}}\)[/tex] be the other leg of this right triangle.
- Using the Pythagorean theorem: [tex]\((20)^2 = (12)^2 + (b_{\text{top}})^2\)[/tex]
[tex]\[
400 = 144 + (b_{\text{top}})^2
\][/tex]
[tex]\[
(b_{\text{top}})^2 = 400 - 144 = 256
\][/tex]
[tex]\[
b_{\text{top}} = \sqrt{256} = 16 \text{ cm}
\][/tex]
4. Form Two Right Triangles from the Bottom Part:
- For the right triangle from the bottom part with the hypotenuse of 13 cm (one side of the kite):
- One leg is half of [tex]\(\overline{EG}\)[/tex] = 12 cm.
- Let [tex]\(b_{\text{bottom}}\)[/tex] be the other leg of this right triangle.
- Using the Pythagorean theorem: [tex]\((13)^2 = (12)^2 + (b_{\text{bottom}})^2\)[/tex]
[tex]\[
169 = 144 + (b_{\text{bottom}})^2
\][/tex]
[tex]\[
(b_{\text{bottom}})^2 = 169 - 144 = 25
\][/tex]
[tex]\[
b_{\text{bottom}} = \sqrt{25} = 5 \text{ cm}
\][/tex]
5. Determine the Full Length of [tex]\(\overline{DF}\)[/tex]:
- The full diagonal [tex]\(\overline{DF}\)[/tex] is formed by adding the two parts:
- [tex]\( \overline{DF} = b_{\text{top}} + b_{\text{bottom}} = 16 \text{ cm} + 5 \text{ cm} = 21 \text{ cm} \)[/tex]
Thus, the length of the other diagonal, [tex]\(\overline{D F}\)[/tex], is 21 cm. Therefore, the correct answer is:
[tex]\[
\boxed{21 \text{ cm}}
\][/tex]
