Question

1. find ab

find bc
ac=ab = 9√2
(right triangle with right angle at a, ac and ab labeled 9√2, bc to be found; handwritten calculations: bc² = ac² + ab², ac² = (9√2)², ab² = (9√2)², bc² = 9√2² + 9√2² = 18 + 18 = 36, bc = √36 = 6)

Explanation:

Step1: Identify Triangle Type

The triangle is right - angled at \(A\), and \(AC = AB=9\sqrt{2}\) (given \(AC = AB\) and \(AB = 9\sqrt{2}\) from the "Find AB" part). So, it is an isosceles right - triangle.

Step2: Apply Pythagorean Theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the Pythagorean theorem is \(c^{2}=a^{2}+b^{2}\). Here, \(a = AC = 9\sqrt{2}\), \(b = AB = 9\sqrt{2}\), and \(c = BC\).
Substitute the values into the formula:

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Step3: Solve for \(BC\)

Take the square root of both sides: \(BC=\sqrt{324} = 18\)

Answer:

\(BC = 18\)