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Intern  Joined: 18 Oct 2009
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Schools: Kellogg
If the vertices of a triangle have coordinates (x,1), (5,1)  [#permalink]

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Question Stats: 50% (02:37) correct 50% (02:33) wrong based on 268 sessions

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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) Angle at the vertex (x,1) is equal to angle at the vertex (5,y)

Originally posted by slingfox on 25 Oct 2009, 02:24.
Last edited by Bunuel on 22 Apr 2012, 02:22, edited 1 time in total.
Edited the question and added the OA
Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: Coordinate Geometry Problem, Triangles  [#permalink]

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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?

Look at the diagram below: Notice that vertex (x,1) will be somewhere on the green line segment and the vertex (5,y) will be somewhere on the blue line segment. So, in any case our triangle will be right angled, with a right angle at vertex (5, 1). Next, the length of the leg on the green line segment will be $$5-x$$ and the length of the leg on the blue line segment will by $$y-1$$. So, the area of the triangle will be: $$area=\frac{1}{2}*(5-x)*(y-1)$$

(1) x=y --> since $$x<5$$ and $$y>1$$ then both x and y are in the range (1,5): $$1<(x=y)<5$$. If we substitute $$y$$ with $$x$$ we'll get: $$area=\frac{1}{2}*(5-x)*(y-1)=\frac{1}{2}*(5-x)*(x-1)$$, different values of $$x$$ give different values for the area (even knowing that $$1<x<5$$). Not sufficient.

(2) Angle at the vertex (x,1) is equal to angle at the vertex (5,y) --> we have an isosceles right triangle: $$5-x=y-1$$. Again if we substitute $$y-1$$ with $$5-x$$ we'll get: $$area=\frac{1}{2}*(5-x)*(y-1)=\frac{1}{2}*(5-x)*(5-x)$$, different values of $$x$$ give different values for the area. Not sufficient.

(1)+(2) $$x=y$$ and $$5-x=y-1$$ --> solve for $$x$$: $$x=y=3$$ --> $$area=\frac{1}{2}*(5-3)*(3-1)=2$$. Sufficient.

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Re: If the vertices of a triangle have coordinates (x,1), (5,1)  [#permalink]

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1) x = y
Even if rational values are acceptable, pick the three allowed integer values (2, 3, 4) and test. You will end up with three different values for the area.

2) Angle at the vertex (x; 1) is equal to agle at the vertex (5; y)
This is telling us that the triangle is a 45-45-90. But we don’t know anything about the length of the two catetuses. At different catetus lengths correspond different area values.

1+2) Sketching down a couple of Cartesian graphics you can figure out that when x = y = 3 the two sides have the same length, that’s what we want → compute the area.

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Re: If the vertices of a triangle have coordinates (x,1), (5,1)  [#permalink]

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_________________ Re: If the vertices of a triangle have coordinates (x,1), (5,1)   [#permalink] 19 Mar 2019, 19:57
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