Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve for the length of the other diagonal, [tex]\( \overline{DF} \)[/tex], in the kite, here's the step-by-step solution:
1. Analyze the Patterns of the Kite:
- We have a kite with one diagonal, [tex]\( \overline{EG} \)[/tex], measuring 24 cm.
- The top two sides of the kite each measure 20 cm, and the bottom two sides each measure 13 cm.
2. Properties of the Kite:
- In a kite, the diagonals intersect each other at right angles (90 degrees).
- The diagonals bisect each other.
3. Divide the Diagonal [tex]\( \overline{EG} \)[/tex]:
- Let's denote the length of [tex]\( \overline{EG} \)[/tex] as 24 cm.
- Since the diagonals bisect each other perpendicularly, half of [tex]\( \overline{EG} \)[/tex] is [tex]\( 24 \, \text{cm} / 2 = 12 \, \text{cm} \)[/tex].
- So each half of [tex]\( \overline{EG} \)[/tex] is 12 cm.
4. Using the Right Triangle Formed by the Diagonals:
- For the top part of the kite, each side measures 20 cm.
- Form a right triangle with one leg as half of [tex]\( \overline{EG} \)[/tex] (12 cm) and the hypotenuse as one of the top sides (20 cm).
5. Applying the Pythagorean Theorem:
- Let’s denote half of the other diagonal [tex]\( \overline{DF} \)[/tex] as [tex]\( x \)[/tex].
- Using the Pythagorean theorem in the triangle formed by the sides of the kite and the halves of the diagonals:
[tex]\[ (\text{side})^2 = (\text{one half of } \overline{DF})^2 + (\text{one half of } \overline{EG})^2 \][/tex]
- For the top side:
[tex]\[ 20^2 = x^2 + 12^2 \][/tex]
- Simplifying:
[tex]\[ 400 = x^2 + 144 \][/tex]
- Isolating [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = 400 - 144 \][/tex]
[tex]\[ x^2 = 256 \][/tex]
- Taking the square root of both sides:
[tex]\[ x = 16 \][/tex]
6. Full Length of [tex]\( \overline{ DF } \)[/tex]:
- Since [tex]\( x \)[/tex] is half of [tex]\( \overline{ DF } \)[/tex], the full length of [tex]\( \overline{ DF } \)[/tex] is:
[tex]\[ \overline{DF} = 2 \times 16 = 32 \, \text{cm} \][/tex]
Therefore, the length of the other diagonal [tex]\( \overline{DF} \)[/tex] is 32 cm.
The correct answer is:
[tex]\[ \boxed{32 \, \text{cm}} \][/tex]
1. Analyze the Patterns of the Kite:
- We have a kite with one diagonal, [tex]\( \overline{EG} \)[/tex], measuring 24 cm.
- The top two sides of the kite each measure 20 cm, and the bottom two sides each measure 13 cm.
2. Properties of the Kite:
- In a kite, the diagonals intersect each other at right angles (90 degrees).
- The diagonals bisect each other.
3. Divide the Diagonal [tex]\( \overline{EG} \)[/tex]:
- Let's denote the length of [tex]\( \overline{EG} \)[/tex] as 24 cm.
- Since the diagonals bisect each other perpendicularly, half of [tex]\( \overline{EG} \)[/tex] is [tex]\( 24 \, \text{cm} / 2 = 12 \, \text{cm} \)[/tex].
- So each half of [tex]\( \overline{EG} \)[/tex] is 12 cm.
4. Using the Right Triangle Formed by the Diagonals:
- For the top part of the kite, each side measures 20 cm.
- Form a right triangle with one leg as half of [tex]\( \overline{EG} \)[/tex] (12 cm) and the hypotenuse as one of the top sides (20 cm).
5. Applying the Pythagorean Theorem:
- Let’s denote half of the other diagonal [tex]\( \overline{DF} \)[/tex] as [tex]\( x \)[/tex].
- Using the Pythagorean theorem in the triangle formed by the sides of the kite and the halves of the diagonals:
[tex]\[ (\text{side})^2 = (\text{one half of } \overline{DF})^2 + (\text{one half of } \overline{EG})^2 \][/tex]
- For the top side:
[tex]\[ 20^2 = x^2 + 12^2 \][/tex]
- Simplifying:
[tex]\[ 400 = x^2 + 144 \][/tex]
- Isolating [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = 400 - 144 \][/tex]
[tex]\[ x^2 = 256 \][/tex]
- Taking the square root of both sides:
[tex]\[ x = 16 \][/tex]
6. Full Length of [tex]\( \overline{ DF } \)[/tex]:
- Since [tex]\( x \)[/tex] is half of [tex]\( \overline{ DF } \)[/tex], the full length of [tex]\( \overline{ DF } \)[/tex] is:
[tex]\[ \overline{DF} = 2 \times 16 = 32 \, \text{cm} \][/tex]
Therefore, the length of the other diagonal [tex]\( \overline{DF} \)[/tex] is 32 cm.
The correct answer is:
[tex]\[ \boxed{32 \, \text{cm}} \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
