To find the length of the other diagonal, [tex]\(\overline{DF}\)[/tex], we can use the properties of a kite and the Pythagorean theorem. Here's a detailed step-by-step solution:
1. Identify the Known Values:
- The length of diagonal [tex]\(\overline{EG}\)[/tex] is 24 cm.
- The top two sides of the kite measure 20 cm each.
- The bottom two sides of the kite measure 13 cm each.
2. Understand the Properties of the Kite:
- Diagonals of a kite intersect at right angles and bisect each other.
3. Divide the Kite:
- When [tex]\(\overline{EG}\)[/tex] intersects [tex]\(\overline{DF}\)[/tex], two right-angled triangles are formed on each side of [tex]\(\overline{EG}\)[/tex].
4. Calculate Half of [tex]\(\overline{EG}\)[/tex]:
- Since [tex]\(\overline{EG}\)[/tex] is bisected, each half is:
[tex]\[
\frac{\overline{EG}}{2} = \frac{24 \text{ cm}}{2} = 12 \text{ cm}
\][/tex]
5. Use the Pythagorean Theorem:
- Consider one of the right-angled triangles with half of [tex]\(\overline{EG}\)[/tex] (which is 12 cm) forming one leg and half of [tex]\(\overline{DF}\)[/tex] forming the other leg (let this be [tex]\(b\)[/tex]). The hypotenuse of this triangle is one of the top sides of the kite, which is 20 cm.
6. Set Up the Pythagorean Theorem:
[tex]\[
(20 \text{ cm})^2 = (12 \text{ cm})^2 + b^2
\][/tex]
7. Solve for [tex]\(b\)[/tex]:
[tex]\[
400 = 144 + b^2
\][/tex]
[tex]\[
b^2 = 400 - 144
\][/tex]
[tex]\[
b^2 = 256
\][/tex]
[tex]\[
b = \sqrt{256}
\][/tex]
[tex]\[
b = 16 \text{ cm}
\][/tex]
8. Calculate the Full Length of [tex]\(\overline{DF}\)[/tex]:
- Since [tex]\(b\)[/tex] represents half of [tex]\(\overline{DF}\)[/tex]:
[tex]\[
\overline{DF} = 2 \times b = 2 \times 16 \text{ cm} = 32 \text{ cm}
\][/tex]
Therefore, the length of the other diagonal [tex]\(\overline{DF}\)[/tex] is 32 cm. The correct answer is:
[tex]\[
\text{32 cm}
\][/tex]
