In the preceding figure, what is the perimeter of ∆ABC inscribed withi

Because the triangle is inscribed in a semicircle, we can see that the radius of the circle is 3 and the diameter is 6. Points A and C will have coordinates (0,3) and (0,-3) respectively. Therefore the long side of the triangle is 6.

The two short sides of the triangle each form a smaller right isosceles triangle. \(\triangle\)AOB and \(\triangle\)COB. The ratio of lengths of a right isosceles triangle are \(1:1:\sqrt{2}\). So the lengths of our triangles are \(3:3:3\sqrt{2}\)

So the total perimeter of the large triangle is \(6+ 2*3\sqrt{2} = 6+6\sqrt {2}\)

Answer D

Alternatively, we could consider the large triangle, which is also an isosceles right triangle with short sides = x and hypotenuse = 6. Then with the same \(1:1:\sqrt{2}\) ratio, our triangle has side lengths x:x:6. \(x\sqrt{2} = 6\)

\(x=\frac{6}{\sqrt{2}} = 3\sqrt{2}\)

Again the perimeter is \(6+ 2*3\sqrt{2} = 6+6\sqrt {2}\)