Question

abcd is a kite, so \overline{ac} \perp \overline{db} and de = eb. calculate the length of \overline{ac}, to the nearest tenth of a centimeter.

Explanation:

Step1: Find \( DE \) using Pythagorean theorem in \( \triangle ADE \)

In right - triangle \( ADE \), by the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \). Let \( AE = 8\) cm and \( AD = 9\) cm. Then \( DE=\sqrt{AD^{2}-AE^{2}}=\sqrt{9^{2}-8^{2}}=\sqrt{81 - 64}=\sqrt{17}\approx4.1\) cm.

Step2: Find \( EC \) using Pythagorean theorem in \( \triangle DEC \)

In right - triangle \( DEC \), \( DE\approx4.1\) cm and \( DC = 10\) cm. By the Pythagorean theorem \( EC=\sqrt{DC^{2}-DE^{2}}=\sqrt{10^{2}-(\sqrt{17})^{2}}=\sqrt{100 - 17}=\sqrt{83}\approx9.1\) cm.

Step3: Calculate \( AC \)

Since \( AC=AE + EC \), and \( AE = 8\) cm, \( EC\approx9.1\) cm. Then \( AC=8 + 9.1=17.1\) cm.

Answer:

\( 17.1\) cm