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Math Expert V
Joined: 02 Sep 2009
Posts: 58344

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Difficulty:   85% (hard)

Question Stats: 46% (02:04) correct 54% (02:05) wrong based on 56 sessions

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Vertices of a triangle have coordinates $$(-2, 2), (3, 2), (x, y)$$. What is the area of the triangle?

(1) $$|y - 2| = 1$$

(2) Angle at the vertex $$(x, y)$$ equals 90 degrees

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Math Expert V
Joined: 02 Sep 2009
Posts: 58344

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Official Solution:

Given two points $$A(-2, 2)$$ and $$B(3, 2)$$. Question asks to find the area of triangle $$ABC$$, where $$C(x,y)$$. Look at the diagram below: (1) $$|y-2|=1$$. Either $$y=3$$ or $$y=1$$, hence vertex $$C$$ could be anywhere on the blue line $$y=3$$ or anywhere on the red line $$y=1$$. But in ANY case the area of $$ABC$$ will be the same: $$area=\frac{1}{2}*base*height$$ so $$base=AB=5$$ and the height would be 1 for any point $$C$$ (see two possible locations of $$C$$: $$C_1$$ and $$C_2$$, the heights of $$ABC_1$$ and $$ABC_2$$ are the same and equal to 1). So, we have that $$area=\frac{1}{2}*base*height=\frac{5}{2}$$. Sufficient.

(2) Angle at the vertex $$(x, y)$$ equals 90 degrees. This statement says that $$ABC$$ is a right triangle with hypotenuse $$AB$$: consider $$AB$$ to be diameter of a circle. In this case $$C$$ could be anywhere on the circle and it will be right angle (if the diameter of the circle is also the inscribed triangle’s side, then that triangle is a right triangle), thus height of $$ABC$$ will be different for different location of point $$C$$, resulting the different areas (see two possible locations of $$C$$: $$C_3$$ and $$C_4$$, heights of $$ABC_3$$ and $$ABC_4$$ are different). Not sufficient.

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Intern  Joined: 02 Feb 2016
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I think this the explanation isn't clear enough, please elaborate. Please explain on this
Math Expert V
Joined: 02 Sep 2009
Posts: 58344

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sricha wrote:
I think this the explanation isn't clear enough, please elaborate. Please explain on this

Please find below link to the good, long discussion on this question:
if-vertices-of-a-triangle-have-coordinates-2-2-3-2-and-82159.html

Hope it helps.
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Manager  Status: Keep it simple stupid!!
Joined: 02 Mar 2016
Posts: 72
Location: India
Concentration: General Management, Operations
GMAT 1: 680 Q47 V36 GPA: 3.37

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I think this is an awesome question and Explanation is superb...!!wow!!
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"Give without remembering, take without forgetting"
- Kudos
Intern  Joined: 29 Jun 2016
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A really good question after a long time
Intern  B
Joined: 13 Oct 2015
Posts: 16

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I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 28 Mar 2017
Posts: 9
GMAT 1: 550 Q43 V23 Show Tags

Very good question. Keep the good work Bunuel.. Thanks for such lovely questions
Manager  G
Joined: 31 Jul 2017
Posts: 88
Location: India
Schools: Anderson '21, LBS '21

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This question needs to be discussed more often.. Just commenting for the same... An awesome question Re: M14-10   [#permalink] 27 Mar 2019, 02:37
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