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Sofia cuts a piece of felt in the shape of a kite for an art project. The top two sides measure 20 cm each and the bottom two sides measure 13 cm each. One diagonal, [tex]\overline{EG}[/tex], measures 24 cm.

What is the length of the other diagonal, [tex]\overline{DF}[/tex]?

A. 5 cm
B. 16 cm
C. 21 cm
D. 32 cm



Answer :

GinnyAnswer
To find the length of the other diagonal [tex]\(\overline{DF}\)[/tex] of the kite, we need to work through the information given about the kite step by step.

1. Identify the given dimensions:
- Top sides of the kite: 20 cm each
- Bottom sides of the kite: 13 cm each
- One diagonal [tex]\(\overline{EG}\)[/tex]: 24 cm

2. Understand the properties of the kite:
- The diagonals of the kite intersect at right angles (90 degrees) and bisect each other.

3. Divide the given diagonal [tex]\(\overline{EG}\)[/tex] into two equal parts:
[tex]\[ \overline{EG} = 24 \text{ cm} \][/tex]
Half of [tex]\(\overline{EG}\)[/tex] would be:
[tex]\[ x = \frac{\overline{EG}}{2} = \frac{24}{2} = 12 \text{ cm} \][/tex]

4. Use the Pythagorean Theorem for the right triangles formed:
- Let's consider one of the right triangles formed by the sides of the kite and half of the diagonal [tex]\(\overline{EG}\)[/tex]:
- For the right triangle with hypotenuse (side of the kite) 20 cm:
[tex]\[ y^2 + x^2 = 20^2 \][/tex]
[tex]\[ y^2 + 12^2 = 400 \][/tex]
[tex]\[ y^2 + 144 = 400 \][/tex]
[tex]\[ y^2 = 400 - 144 = 256 \][/tex]
[tex]\[ y = \sqrt{256} = 16 \text{ cm} \][/tex]

5. Determine one half of the other diagonal:
[tex]\[ y = 16 \text{ cm} \][/tex]
Since diagonals bisect each other at right angles, and we need the full length of [tex]\(\overline{DF}\)[/tex]:
[tex]\[ \overline{DF} = 2 \times y = 2 \times 16 = 32 \text{ cm} \][/tex]

Therefore, the length of the other diagonal [tex]\(\overline{DF}\)[/tex] is:
[tex]\[ \boxed{32 \text{ cm}} \][/tex]

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