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Sagot :
Certainly! Let's solve the problem step-by-step. Sofia has a piece of felt cut in the shape of a kite. The kite has the following measurements:
- Top two sides are 20 cm each.
- Bottom two sides are 13 cm each.
- One diagonal, [tex]\( \overline{EG} \)[/tex], measures 24 cm.
We need to find the length of the other diagonal.
1. Understand the Kite Properties:
- A kite has two pairs of adjacent sides that are equal.
- The diagonals of a kite are perpendicular bisectors of each other where they intersect.
2. Use the Given Information:
- Denote the diagonals of the kite by [tex]\( \overline{EG} \)[/tex] and the other diagonal [tex]\( \overline{FH} \)[/tex].
- Given [tex]\( \overline{EG} = 24 \)[/tex] cm, it means each half of this diagonal (since diagonals bisect each other at right angles) is [tex]\( \frac{24}{2} = 12 \)[/tex] cm.
3. Use Pythagoras' Theorem in Right Triangles Formed:
- Consider right triangles formed by the intersection of the diagonals.
- Let's divide the kite into two right triangles by the other diagonal [tex]\( \overline{FH} \)[/tex].
4. Top Side Triangle Calculation:
- Observe the top side forms two right triangles each with one of the halves of the other diagonal ([tex]\( \frac{\overline{FH}}{2} \)[/tex]).
- For the top two sides (each 20 cm), the right triangle with the half diagonal of [tex]\( \overline{EG} \)[/tex] (12 cm):
[tex]\[ 20^2 = 12^2 + \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ 400 = 144 + \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ 256 = \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ \frac{\overline{FH}}{2} = \sqrt{256} = 16 \text{ cm} \][/tex]
5. Bottom Side Triangle Calculation:
- For the bottom two sides (each 13 cm), the right triangle with the half of the other diagonal [tex]\( \frac{\overline{FH}}{2} \)[/tex] remains (12 cm):
[tex]\[ 13^2 = 12^2 + \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ 169 = 144 + \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ 25 = \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ \frac{\overline{FH}}{2} = \sqrt{25} = 5 \text{ cm} \][/tex]
6. Combine the Calculations:
- Since [tex]\( \overline{FH} \)[/tex] is the other diagonal:
[tex]\[ \overline{FH} = 2 \times 16 + 2 \times 5 = 16 + 5 = 21 \text{ cm} \][/tex]
So, the length of the other diagonal [tex]\( \overline{FH} \)[/tex] is:
```
21 cm
```
Thus, the answer is 21 cm.
- Top two sides are 20 cm each.
- Bottom two sides are 13 cm each.
- One diagonal, [tex]\( \overline{EG} \)[/tex], measures 24 cm.
We need to find the length of the other diagonal.
1. Understand the Kite Properties:
- A kite has two pairs of adjacent sides that are equal.
- The diagonals of a kite are perpendicular bisectors of each other where they intersect.
2. Use the Given Information:
- Denote the diagonals of the kite by [tex]\( \overline{EG} \)[/tex] and the other diagonal [tex]\( \overline{FH} \)[/tex].
- Given [tex]\( \overline{EG} = 24 \)[/tex] cm, it means each half of this diagonal (since diagonals bisect each other at right angles) is [tex]\( \frac{24}{2} = 12 \)[/tex] cm.
3. Use Pythagoras' Theorem in Right Triangles Formed:
- Consider right triangles formed by the intersection of the diagonals.
- Let's divide the kite into two right triangles by the other diagonal [tex]\( \overline{FH} \)[/tex].
4. Top Side Triangle Calculation:
- Observe the top side forms two right triangles each with one of the halves of the other diagonal ([tex]\( \frac{\overline{FH}}{2} \)[/tex]).
- For the top two sides (each 20 cm), the right triangle with the half diagonal of [tex]\( \overline{EG} \)[/tex] (12 cm):
[tex]\[ 20^2 = 12^2 + \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ 400 = 144 + \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ 256 = \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ \frac{\overline{FH}}{2} = \sqrt{256} = 16 \text{ cm} \][/tex]
5. Bottom Side Triangle Calculation:
- For the bottom two sides (each 13 cm), the right triangle with the half of the other diagonal [tex]\( \frac{\overline{FH}}{2} \)[/tex] remains (12 cm):
[tex]\[ 13^2 = 12^2 + \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ 169 = 144 + \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ 25 = \left( \frac{\overline{FH}}{2} \right)^2 \][/tex]
[tex]\[ \frac{\overline{FH}}{2} = \sqrt{25} = 5 \text{ cm} \][/tex]
6. Combine the Calculations:
- Since [tex]\( \overline{FH} \)[/tex] is the other diagonal:
[tex]\[ \overline{FH} = 2 \times 16 + 2 \times 5 = 16 + 5 = 21 \text{ cm} \][/tex]
So, the length of the other diagonal [tex]\( \overline{FH} \)[/tex] is:
```
21 cm
```
Thus, the answer is 21 cm.
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