To find the length of the other diagonal [tex]\( \overline{DF} \)[/tex] of the kite, we can use the properties of kites and the given information. Let's go through the steps:
1. Understanding the Problem:
- We have a kite with the top two sides measuring 20 cm each and the bottom two sides measuring 13 cm each.
- One diagonal [tex]\( \overline{EG} \)[/tex] measures 24 cm.
2. Find the Areas of Triangles Formed by the Diagonal [tex]\( \overline{EG} \)[/tex]:
- Since the kite is symmetric, the diagonal [tex]\( \overline{EG} \)[/tex] splits the kite into two pairs of congruent right triangles.
- We can use half of [tex]\( \overline{EG} \)[/tex] (which is [tex]\( \frac{24}{2} = 12 \)[/tex] cm) as the height in these right triangles.
3. Calculate the Area of the Kite by Considering the Triangles:
- The kite consists of two pairs of congruent right triangles.
- For the top pair of right triangles, with each having a base of 20 cm and a height of 12 cm:
[tex]\[
\text{Area of one triangle (top)} = \frac{1}{2} \times 20 \times 12 = 120 \, \text{cm}^2
\][/tex]
- For the bottom pair of right triangles, with each having a base of 13 cm and a height of 12 cm:
[tex]\[
\text{Area of one triangle (bottom)} = \frac{1}{2} \times 13 \times 12 = 78 \, \text{cm}^2
\][/tex]
- Since there are two of each pair of triangles, their combined areas are:
[tex]\[
\text{Total area of the kite} = 2 \times (120 + 78) = 2 \times 198 = 396 \, \text{cm}^2
\][/tex]
4. Using the Formula for Area of a Kite:
- The area of a kite can also be calculated using its diagonals:
[tex]\[
\text{Area} = \frac{1}{2} \times D_1 \times D_2
\][/tex]
- We know the area (396 cm²) and one diagonal [tex]\( D_1 = 24 \)[/tex] cm.
- We need to find the other diagonal [tex]\( D_2 \)[/tex].
5. Solve for [tex]\( D_2 \)[/tex]:
[tex]\[
396 = \frac{1}{2} \times 24 \times D_2
\][/tex]
[tex]\[
396 = 12 \times D_2
\][/tex]
[tex]\[
D_2 = \frac{396}{12} = 33 \, \text{cm}
\][/tex]
Therefore, the length of the other diagonal [tex]\( \overline{DF} \)[/tex] is [tex]\( 33 \)[/tex] cm.
