Question

type the correct answer in the box
find the length of \\( \overline { s u } \\) in the isosceles trapezoid.

\\( \

$$\begin{array} { l } { s w = 1.7 } \\\\ { v w = 3 } \\\\ { s v = 1.5 } \\\\ { t u = 1.5 } \\\\ { s u = \\square } \\end{array}$$

\\)

Explanation:

Step1: Prove $\triangle SVW\cong\triangle TUW$

In an isosceles trapezoid \(SV = TU = 1.5\), \(\angle SVW=\angle TUW\) (alternate - interior angles as \(SV\parallel TU\)), \(\angle SWV=\angle T WU\) (vertical angles). By \(AAS\) (Angle - Angle - Side) congruence criterion, \(\triangle SVW\cong\triangle TUW\). Then \(SW = TW = 1.7\), \(VW = UW = 3\).

Step2: Use the Pythagorean theorem in \(\triangle SWU\)

In right - triangle \(SWU\) (we can find it's a right - triangle by using the properties of congruent triangles and trapezoid, or by calculating side lengths and using the converse of the Pythagorean theorem. Another way: since \(\triangle SVW\cong\triangle TUW\), and in an isosceles trapezoid, the diagonals \(SU\) and \(VT\) intersect at \(W\). The formula for the length of \(SU\) using the segments \(SW\) and \(UW\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = SW = 1.7\), \(b=UW = 3\), and \(c = SU\).

$$SU=\sqrt{SW^{2}+UW^{2}}$$

$$SU=\sqrt{1.7^{2}+3^{2}}=\sqrt{2.89 + 9}=\sqrt{11.89}\approx3.45$$

Answer:

\(3.45\)