To find the length of the other diagonal, [tex]\(\overline{DF}\)[/tex], in the kite given that one diagonal, [tex]\(\overline{EG}\)[/tex], measures 24 cm, we can use the following steps:
1. Notice the Symmetry and Properties of the Kite:
Given that Sofia's kite has the top two sides measuring 20 cm each and the bottom two sides measuring 13 cm each, we can split the kite into two congruent triangles along one of its diagonals.
2. Using Known Formulas:
The area of a kite can be determined using its diagonals with the formula:
[tex]\[
\text{Area} = \frac{1}{2} \times d_1 \times d_2
\][/tex]
where [tex]\(d_1\)[/tex] and [tex]\(d_2\)[/tex] are the lengths of the two diagonals.
3. Calculate the Area Using Side Lengths:
Another way to determine the kite's area is by recognizing it's made up of two pairs of congruent triangles. Each of these right triangles can be analyzed by its base and height. Given the kite is symmetric, we can compute the area using both sets of side lengths. The area for each set of congruent triangles is:
[tex]\[
\text{Area of each pair of triangles} = \frac{1}{2} \times \text{side1} \times \text{other side}
\][/tex]
4. Area Calculation:
Since we know one diagonal [tex]\(\overline{EG}\)[/tex], let’s divide the kite into right-angled triangles using the known top and bottom sides:
[tex]\[
\text{Area of kite} = 2 \times \left( \frac{1}{2} \times 20 \times 13 \right)
\][/tex]
Simplifying, we get:
[tex]\[
\text{Area} = 2 \times 130 = 260 \text{ square cm}
\][/tex]
5. Using the Diagonal to Find Missing Diagonal:
The total area, [tex]\( \frac{1}{2} \times 24 \times \text{DF} \)[/tex], must also be equal to the total area we already calculated. So, we set up the equation:
[tex]\[
\frac{1}{2} \times 24 \times \text{DF} = 260
\][/tex]
Solving for DF:
[tex]\[
12 \times \text{DF} = 260
\][/tex]
[tex]\[
\text{DF} = \frac{260}{12}
\][/tex]
Simplifying further:
[tex]\[
\text{DF} \approx 21.67
\][/tex]
Rounded to the nearest integer, the length of the other diagonal [tex]\(\overline{DF}\)[/tex] is 22 cm.
The correct length of the other diagonal [tex]\(\overline{DF}\)[/tex] is closest to 22 cm (not listed in the options), but among the options given, the correct approximate value is:
21 cm
