To determine the length of the other diagonal ([tex]\(\overline{DF}\)[/tex]) in the kite, follow these steps:
1. Understand Kite Properties:
- In a kite, the diagonals are perpendicular (they intersect at right angles).
- The diagonals bisect each other.
2. Assign the given values:
- The length of diagonal [tex]\(\overline{EG}\)[/tex] is 24 cm.
- The top sides of the kite (legs of the kite) each measure 20 cm.
- The bottom sides each measure 13 cm.
3. Split the diagonals:
Since diagonals intersect perpendicularly and bisect each other, each half of diagonal [tex]\(\overline{EG}\)[/tex] will be:
[tex]\[
\overline{EG} = 24 \text{ cm} \quad \Rightarrow \quad \frac{\overline{EG}}{2} = 12 \text{ cm}
\][/tex]
4. Form Right Triangles:
At the intersection of diagonals, four right-angled triangles are formed. Focus on one of the right-angled triangles that uses a 20 cm side, half of [tex]\(\overline{EG}\)[/tex] which is 12 cm, and half of the unknown diagonal ([tex]\(\overline{DF}\)[/tex]).
5. Using Pythagorean Theorem:
Let's denote half of [tex]\(\overline{DF}\)[/tex] as [tex]\(x\)[/tex]. Therefore, in a right triangle where:
- One leg is 12 cm (half of [tex]\(\overline{EG}\)[/tex])
- The hypotenuse is 20 cm (side of the kite)
The theorem states:
[tex]\[
20^2 = 12^2 + x^2
\][/tex]
[tex]\[
400 = 144 + x^2
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x^2 = 400 - 144
\][/tex]
[tex]\[
x^2 = 256
\][/tex]
[tex]\[
x = \sqrt{256}
\][/tex]
[tex]\[
x = 16
\][/tex]
6. Determine Full Length of [tex]\(\overline{DF}\)[/tex]:
Since [tex]\(x\)[/tex] represents half of [tex]\(\overline{DF}\)[/tex]:
[tex]\[
\overline{DF} = 2 \times x = 2 \times 16 = 32 \text{ cm}
\][/tex]
Therefore, the length of the other diagonal, [tex]\(\overline{DF}\)[/tex], is 32 cm.
