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16 Sep 2014, 01:05
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A
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If vertices of a triangle have coordinates \((-1, 1)\), \((4, 1)\), and \((x, y)\), what is the area of the triangle?
(1) \(y^2 - 2y - 3 = 0\)
(2) \(x^2 = y^2\)
(1) \(y^2 - 2y - 3 = 0\)
(2) \(x^2 = y^2\)
16 Sep 2014, 01:05
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Official Solution:
If vertices of a triangle have coordinates \((-1, 1)\), \((4, 1)\), and \((x, y)\), what is the area of the triangle?
(1) \(y^2-2y-3=0\).
Solving this equation, we get \(y=3\) or \(y=-1\). Consider the diagram below:
The third vertex is either on the line \(y=3\) (blue line) or on the line \(y=-1\) (green line). Observe that in either case, the height of the triangle is 2 (based on the two possible positions of the third vertex shown in the diagram). Since the length of the base of the triangle is \(4-(-1)=5\), the area can be calculated as \(\frac{1}{2}*base *height = \frac{1}{2}*5*2 = 5\). Sufficient.
(2) \(x^2=y^2\).
This implies that \(|x|=|y|\). However, this statement alone is clearly insufficient to determine the area of the triangle, as various possibilities for \((x, y)\) exist, such as \((x, y) = (3, 3)\) or \((x, y) = (5, 5)\).
Answer: A
If vertices of a triangle have coordinates \((-1, 1)\), \((4, 1)\), and \((x, y)\), what is the area of the triangle?
(1) \(y^2-2y-3=0\).
Solving this equation, we get \(y=3\) or \(y=-1\). Consider the diagram below:
The third vertex is either on the line \(y=3\) (blue line) or on the line \(y=-1\) (green line). Observe that in either case, the height of the triangle is 2 (based on the two possible positions of the third vertex shown in the diagram). Since the length of the base of the triangle is \(4-(-1)=5\), the area can be calculated as \(\frac{1}{2}*base *height = \frac{1}{2}*5*2 = 5\). Sufficient.
(2) \(x^2=y^2\).
This implies that \(|x|=|y|\). However, this statement alone is clearly insufficient to determine the area of the triangle, as various possibilities for \((x, y)\) exist, such as \((x, y) = (3, 3)\) or \((x, y) = (5, 5)\).
Answer: A
23 Jul 2016, 04:39
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I think this is a high-quality question and I agree with explanation.
