Question

in the diagram, ( kl = 2sqrt{2} ), ( lm = sqrt{5} ), and ( mn = sqrt{2} ). what is the perimeter of isosceles trapezoid klmn?
( sqrt{2} ) units
( sqrt{5} ) units
( 3sqrt{2}+2sqrt{5} ) units
( 4sqrt{2}+2sqrt{5} ) units

Explanation:

Step1: Recall the property of isosceles trapezoid

In an isosceles trapezoid, non - parallel sides are equal. Here, \(KL = KN\) and \(LM=MN\). Given \(KL = 2\sqrt{2}\), \(LM=\sqrt{5}\), \(MN=\sqrt{2}\).

Step2: Calculate the perimeter formula

The perimeter \(P\) of a trapezoid \(KLMN\) is \(P=KL + LM+MN + KN\).
Since \(KL = KN = 2\sqrt{2}\) and \(LM = MN=\sqrt{5}\).
Substitute the values into the formula:
\(P=(2\sqrt{2})+\sqrt{5}+\sqrt{2}+(2\sqrt{2})\)

Step3: Combine like terms

Combine the terms with \(\sqrt{2}\): \((2\sqrt{2}+\sqrt{2}+2\sqrt{2})+( \sqrt{5}+\sqrt{5})\)
Using the rule \(a\sqrt{b}+c\sqrt{b}=(a + c)\sqrt{b}\), we have \((2 + 1+2)\sqrt{2}+(1 + 1)\sqrt{5}\)
\(P = 4\sqrt{2}+2\sqrt{5}\)

Answer:

\(4\sqrt{2}+2\sqrt{5}\text{ units}\) (the fourth option)