To find the length of the other diagonal, [tex]\(\overline{DF}\)[/tex], in the kite, we need to use the properties of kites and the Pythagorean theorem. Here are the detailed steps:
1. Understanding the Problem:
- A kite has two pairs of adjacent sides that are equal.
- The top two sides measure 20 cm each.
- The bottom two sides measure 13 cm each.
- One diagonal, [tex]\(\overline{EG}\)[/tex], measures 24 cm.
- We need to find the length of the other diagonal, [tex]\(\overline{DF}\)[/tex].
2. Properties of Kites:
- The diagonals of a kite are perpendicular bisectors of each other.
- This means they intersect at right angles and each diagonal cuts the other into two equal parts.
3. Breaking Down the Kite:
- Let [tex]\(O\)[/tex] be the point where the diagonals intersect.
- Therefore, each half of the diagonal [tex]\(\overline{EG}\)[/tex], which measures 24 cm, will measure 12 cm. So we have:
[tex]\[
EO = OG = 12 \, \text{cm}
\][/tex]
4. Forming Two Right Triangles:
- Consider the right triangle [tex]\(AEO\)[/tex] (or [tex]\(BEO\)[/tex] since both are symmetrical).
- In this right triangle, [tex]\(AE = 20 \, \text{cm}\)[/tex] and [tex]\(EO = 12 \, \text{cm}\)[/tex].
5. Applying the Pythagorean Theorem:
- Using the Pythagorean theorem [tex]\(a^2 + b^2 = c^2\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the legs and [tex]\(c\)[/tex] is the hypotenuse:
[tex]\[
AO^2 + EO^2 = 20^2
\][/tex]
- We substitute the known values:
[tex]\[
AO^2 + 12^2 = 20^2
\][/tex]
[tex]\[
AO^2 + 144 = 400
\][/tex]
[tex]\[
AO^2 = 400 - 144
\][/tex]
[tex]\[
AO^2 = 256
\][/tex]
[tex]\[
AO = \sqrt{256}
\][/tex]
[tex]\[
AO = 16 \, \text{cm}
\][/tex]
- Since [tex]\(AO\)[/tex] is half of the diagonal [tex]\(\overline{DF}\)[/tex], the full diagonal [tex]\(\overline{DF}\)[/tex] will be:
[tex]\[
DF = 2 \times AO = 2 \times 16 = 32 \, \text{cm}
\][/tex]
Hence, the length of the other diagonal [tex]\(\overline{DF}\)[/tex] is [tex]\(32 \, \text{cm}\)[/tex]. The correct answer is:
[tex]\[
\boxed{32 \, \text{cm}}
\][/tex]
