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16 Sep 2014, 00:36
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If the vertices of a triangle have coordinates \((x, 1)\), \((5, 1)\), and \((5, y)\) where \(x < 5\) and \(y > 1\), what is the area of the triangle?
(1) \(x = y\).
(2) The angle at the vertex \((x, 1)\) is equal to the angle at the vertex \((5, y)\).
(1) \(x = y\).
(2) The angle at the vertex \((x, 1)\) is equal to the angle at the vertex \((5, y)\).
16 Sep 2014, 00:36
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Official Solution:
If the vertices of a triangle have coordinates \((x, 1)\), \((5, 1)\), and \((5, y)\) where \(x < 5\) and \(y > 1\), what is the area of the triangle?
Consider the diagram below:
Observe that the vertex \((x, 1)\) will be located somewhere along the green line segment, and the vertex \((5, y)\) will be situated along the blue line segment. In all cases, the triangle will have a right angle at vertex \((5, 1)\). The length of the leg along the green line segment is \(5-x\), and the length of the leg along the blue line segment is \(y-1\). Thus, the area of the triangle is given by: \(\text{area}=\frac{1}{2}*(5-x)*(y-1)\).
(1) \(x=y\).
Since \(x < 5\) and \(y > 1\), both \(x\) and \(y\) lie in the range (1, 5): \(1 < (x=y) < 5\). Substituting \(y\) with \(x\), we obtain: \(\text{area}=\frac{1}{2}*(5-x)*(y-1)=\frac{1}{2}*(5-x)*(x-1)\). Different values of \(x\) within the given range yield different area values. This statement is not sufficient.
(2) The angle at the vertex \((x, 1)\) is equal to the angle at the vertex \((5, y)\).
This condition implies that the triangle is an isosceles right triangle: \(5-x=y-1\). Substituting \(y-1\) with \(5-x\), we get: \(\text{area}=\frac{1}{2}*(5-x)*(y-1)=\frac{1}{2}*(5-x)*(5-x)\). Different values of \(x\) yield different area values. This statement is not sufficient.
(1)+(2) Combining both statements, we have \(x=y\) and \(5-x=y-1\). Solving gives \(x=y=3\). Thus, the area of the triangle is \(\text{area}=\frac{1}{2}*(5-3)*(3-1)=2\). Sufficient.
Answer: C
If the vertices of a triangle have coordinates \((x, 1)\), \((5, 1)\), and \((5, y)\) where \(x < 5\) and \(y > 1\), what is the area of the triangle?
Consider the diagram below:
Observe that the vertex \((x, 1)\) will be located somewhere along the green line segment, and the vertex \((5, y)\) will be situated along the blue line segment. In all cases, the triangle will have a right angle at vertex \((5, 1)\). The length of the leg along the green line segment is \(5-x\), and the length of the leg along the blue line segment is \(y-1\). Thus, the area of the triangle is given by: \(\text{area}=\frac{1}{2}*(5-x)*(y-1)\).
(1) \(x=y\).
Since \(x < 5\) and \(y > 1\), both \(x\) and \(y\) lie in the range (1, 5): \(1 < (x=y) < 5\). Substituting \(y\) with \(x\), we obtain: \(\text{area}=\frac{1}{2}*(5-x)*(y-1)=\frac{1}{2}*(5-x)*(x-1)\). Different values of \(x\) within the given range yield different area values. This statement is not sufficient.
(2) The angle at the vertex \((x, 1)\) is equal to the angle at the vertex \((5, y)\).
This condition implies that the triangle is an isosceles right triangle: \(5-x=y-1\). Substituting \(y-1\) with \(5-x\), we get: \(\text{area}=\frac{1}{2}*(5-x)*(y-1)=\frac{1}{2}*(5-x)*(5-x)\). Different values of \(x\) yield different area values. This statement is not sufficient.
(1)+(2) Combining both statements, we have \(x=y\) and \(5-x=y-1\). Solving gives \(x=y=3\). Thus, the area of the triangle is \(\text{area}=\frac{1}{2}*(5-3)*(3-1)=2\). Sufficient.
Answer: C
13 Dec 2019, 07:55
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And here's a video explanation. Enjoy!
