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Since the area of the triangle \(OBC\) is \(8\) and the length of the base \(OC\) is \(4\), and the area of a triangle = \(\frac{1}{2}bh\), we have:
\(8 = \frac{1}{2}(4)h\)
\(8 = 2h\)
\(h = 4.\)
The height of the triangle is \(4\), and the y-coordinate of point \(B\) is \(4.\)
Since point \(B\) is on the line \(y = 2x\), we have \(4 = 2x\) or \(x = 2.\)
Then we have point \(B(2, 4).\)
We can then determine the equation of the line passing through \(B (2, 4)\) and \(C (4, 0).\) We first determine the slope of the line using the equation
(y
1 - y
2) / (x
1 - x
2)
\(\frac{(4 - 0) }{ (2 - 4)}\)
\(\frac{4 }{ -2}\)
Slope = \(m = -2.\)
Then, using the point \((4, 0)\) and \(m = 4\) we get:
\(y = mx + b\)
\(0 = -2(4) + b\)
\(b = 8\)
The line passing through \(B\) and \(C\) is \(y = -2x + 8.\)
Point \(A\) has an x-coordinate of \(0\), so \(y = -2(0) + 9\), or \(y = 8.\)
Then we have point \(A(0,8).\)
The triangle \(AOB\) has base \(OA = 8\), and the height equals \(2\) since the x-coordinate of \(B\) is \(2.\) Then the area of triangle \(OAB\) is \((\frac{1}{2})*8*2 = 8.\)
Therefore, C is the answer.
Answer: C
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