Question

ABCD is a rectangle with its vertices at (2, --2), (8, 4), (4, 8) and (--2, 2) taken in order. Length of its diagonal is

• A. \(4\sqrt{2}\)
• B. \(6\sqrt{2}\)
• C. \(4\sqrt{26}\)
• D. \(2\sqrt{26}\)

Answer 1

The Correct Option is D

Given:
Vertices of rectangle:
\(A(2, -2)\), \(B(8, 4)\), \(C(4, 8)\), \(D(-2, 2)\)

Step 1: Use distance formula to find diagonal \(AC\)
Distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Coordinates:
\(A = (2, -2)\), \(C = (4, 8)\)
Calculate:
\[ AC = \sqrt{(4 - 2)^2 + (8 - (-2))^2} = \sqrt{2^2 + 10^2} = \sqrt{4 + 100} = \sqrt{104} \]

Step 2: Simplify \(\sqrt{104}\)
Prime factorization:
\[ 104 = 4 \times 26 \] \[ AC = \sqrt{4 \times 26} = \sqrt{4} \times \sqrt{26} = 2 \sqrt{26} \]

Step 3: Use distance formula to find diagonal \(BD\)
Coordinates:
\(B = (8, 4)\), \(D = (-2, 2)\)
Calculate:
\[ BD = \sqrt{(-2 - 8)^2 + (2 - 4)^2} = \sqrt{(-10)^2 + (-2)^2} = \sqrt{100 + 4} = \sqrt{104} = 2 \sqrt{26} \]

Final Answer:
Length of the diagonal is:
\[ \boxed{2 \sqrt{26}} \]