Question

find the length of \\(\overline{be}\\) for rectangle abcd.
the length of \\(\overline{be}\\) is \\(\square\\).
(type an integer or decimal rounded to one decimal place as needed.)

Explanation:

Step1: Recall rectangle diagonal properties

In a rectangle, diagonals are equal and bisect each other. So, \( AC = BD \) and \( BE=\frac{1}{2}BD \). First, find the length of diagonal \( BD \) using the Pythagorean theorem. The sides of the rectangle are \( BC = 5 \) and \( CD = 7 \) (since \( ABCD \) is a rectangle, \( AB = CD = 7 \), \( BC = AD = 5 \)). For right triangle \( BCD \), \( BD^{2}=BC^{2}+CD^{2} \).

Step2: Calculate diagonal \( BD \)

Substitute \( BC = 5 \) and \( CD = 7 \) into the formula: \( BD^{2}=5^{2}+7^{2}=25 + 49=74 \), so \( BD=\sqrt{74}\approx8.6 \).

Step3: Find length of \( BE \)

Since \( E \) is the midpoint of \( BD \) (diagonals bisect each other in a rectangle), \( BE=\frac{1}{2}BD \). Substitute \( BD\approx8.6 \): \( BE=\frac{1}{2}\times8.6 = 4.3 \).

Answer:

\( 4.3 \)