Question

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{eg}), measures 24 cm. what is the length of the other diagonal, (overline{df})? 5 cm 16 cm 21 cm 32 cm

Explanation:

Step1: Use the property of kite's diagonals

The diagonals of a kite are perpendicular to each other. Let the intersection point of the diagonals \(EG\) and \(DF\) be \(O\). Since \(EG = 24\) cm, then \(EO=OG=\frac{EG}{2}=12\) cm.

Step2: Apply the Pythagorean theorem in right - triangles

In right - triangle \(FEO\), using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 20\) cm (side \(FE\)) and \(a = 12\) cm (\(EO\)). Let \(FO\) be \(x\), then \(x=\sqrt{20^{2}-12^{2}}=\sqrt{(20 + 12)(20 - 12)}=\sqrt{32\times8}=\sqrt{256}=16\) cm.
In right - triangle \(DEO\), using the Pythagorean theorem, where \(c = 13\) cm (side \(DE\)) and \(a = 12\) cm (\(EO\)). Let \(DO\) be \(y\), then \(y=\sqrt{13^{2}-12^{2}}=\sqrt{(13 + 12)(13 - 12)}=\sqrt{25\times1}=5\) cm.

Step3: Calculate the length of \(DF\)

Since \(DF=FO + OD\), and \(FO = 16\) cm, \(OD = 5\) cm, then \(DF=16 + 5=21\) cm.

Answer:

21 cm