Question

in the isosceles right triangle above, \\( \overline { a c } = 8 \\) inches. what is the length, in inches, of \\( \overline { a b } \\)?

Explanation:

Step1: Identify the properties of an isosceles right - triangle

In an isosceles right - triangle \( \triangle ABC\) with \(\angle C = 90^{\circ}\), \(AC=BC = 8\) inches (since it's isosceles).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a,b\) are the legs. Here \(a = AC\), \(b = BC\) and \(c=AB\).
Substitute \(a = 8\) and \(b = 8\) into the formula:
\(AB^{2}=AC^{2}+BC^{2}\)
\(AB^{2}=8^{2}+8^{2}\)
\(AB^{2}=64 + 64\)
\(AB^{2}=128\)
\(AB=\sqrt{128}\)
Simplify \(\sqrt{128}=\sqrt{64\times2}\)
Since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 64\), \(b = 2\)), then \(AB=\sqrt{64}\times\sqrt{2}\)
\(AB = 8\sqrt{2}\)

Answer:

D. \(8\sqrt{2}\)