Question

abcd is a kite, so \\( \overline { a c } \perp \overline { d b } \\) and \\( d e = e b \\). calculate the length of \\( \overline { a c } \\), to the nearest tenth of a centimeter.
answer attempt 1 out of 2
\\( a c \approx \\) cm submit answer

Explanation:

Step1: Find the length of \( DE \) and \( EB \)

Since \( DE = EB \) and \( DB=10\) cm, then \( DE=\frac{DB}{2}=\frac{10}{2} = 5\) cm.

Step2: Use the Pythagorean theorem to find \( CE \)

In right - triangle \( CDE \), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c = CD = 8\) cm and \(a = DE = 5\) cm, and \(b = CE\)). So \(CE=\sqrt{CD^{2}-DE^{2}}=\sqrt{8^{2}-5^{2}}=\sqrt{64 - 25}=\sqrt{39}\approx6.2\) cm.

Step3: Use the Pythagorean theorem to find \( AE \)

In right - triangle \( ADE \), by the Pythagorean theorem (\(c = AD = 9\) cm and \(a = DE = 5\) cm, and \(b = AE\)). So \(AE=\sqrt{AD^{2}-DE^{2}}=\sqrt{9^{2}-5^{2}}=\sqrt{81 - 25}=\sqrt{56}\approx7.5\) cm.

Step4: Calculate the length of \( AC \)

Since \(AC=CE + AE\), then \(AC=\sqrt{39}+\sqrt{56}\approx6.2+7.5 = 13.7\) cm.

Answer:

\(13.7\)