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, $\overline{E G}$, measures 24 cm.
What is the length of the other diagonal, $\overline{ DF }$?
A. 5 cm
B. 16 cm
C. 21 cm
D. 32 cm
Answer (2)
Recognize that the diagonals of a kite are perpendicular, and one diagonal bisects the other.
Apply the Pythagorean theorem to find the lengths of the segments of the other diagonal.
Sum the lengths of the segments to find the total length of the diagonal.
Conclude that the length of the diagonal D F is 21 cm.
Explanation
Analyze the given information and properties of a kite. Let the kite be D EFG , where D E = D F = 20 cm and EG = FG = 13 cm. Let EG = 24 cm. The diagonals of a kite are perpendicular. Let the intersection of the diagonals be H . Since the diagonal D F is the axis of symmetry, E H = H G = 2 24 = 12 cm.
Apply the Pythagorean theorem to triangle DEH. Consider triangle D E H , which is a right triangle. Use the Pythagorean theorem to find DH : D E 2 = E H 2 + D H 2 , so 2 0 2 = 1 2 2 + D H 2 . Thus, D H 2 = 400 − 144 = 256 , and DH = 256 = 16 cm.
Apply the Pythagorean theorem to triangle EHF. Consider triangle E H F , which is a right triangle. Use the Pythagorean theorem to find H F : E F 2 = E H 2 + H F 2 , so 1 3 2 = 1 2 2 + H F 2 . Thus, H F 2 = 169 − 144 = 25 , and H F = 25 = 5 cm.
Calculate the length of diagonal DF. The length of the diagonal D F is DH + H F = 16 + 5 = 21 cm.
Examples
Kites are not just fun toys; they're also a great example of geometry in action! The symmetrical shape of a kite, with its two pairs of equal-length sides and perpendicular diagonals, makes it a perfect real-world illustration of geometric principles. Understanding these principles allows us to calculate dimensions, angles, and areas, which is useful in various fields like architecture, engineering, and even art. For instance, architects might use kite geometry to design unique window shapes or roof structures, while artists could incorporate kite-like patterns into their designs for aesthetic appeal.
The length of diagonal D F in the kite is determined to be 21 cm by applying the Pythagorean theorem to the right triangles formed by the diagonals. The individual segments of diagonal D F were calculated as 16 cm and 5 cm, which summed to give the final answer. Therefore, the correct option is C. 21 cm.
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