M08-06

Question

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) .

Bunuel · Accepted Answer

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

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GMATNinja · Answer

And here's a video explanation. Enjoy!

https://www.youtube.com/watch?v=VOCyhtnUu2I

WillGetIt · Answer

Hello Bunuel,

I have a concern regarding this question. I think OA should be E not C.

How can x=y translates into 5−x=y−1 ??

Please see the attached pic, using which we can see that even when X = Y areas could be different.

Please assist.

Thanks

Bunuel · Answer

In your example with red hypotenuse how is it true that x=y?...

Senthil7 · Answer

I think this is a high-quality question and I agree with explanation.

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Bunuel · Answer

Similar questions: in-the-diagram-above-coordinates-are-given-for-three-of-the-vertices-194197.html if-vertices-of-a-triangle-have-coordinates-87344.html if-vertices-of-a-triangle-have-coordinates-2-2-3-2-and-82159.html

KarishmaB · Answer

Responding to a pm:

Angle at (5, 1) will be 90 degrees.

(1)  x = y 
The co-ordinates are (x, 1), (5, 1) and (5, x). 
Not sufficient alone since we don't know x.

(2) Angle at the vertex  (x, 1)  is equal to angle at the vertex  (5, y)  
This means the two sides are equal. So 
5 - x = y - 1
Not sufficient alone.

Using both statements,

5 - x = x - 1
x = 3 = y

We get all co-ordinates so we will be able to find the area. 
Answer (C)

tejasa07 · Answer

In this case, from the problem statement itself we can deduce that x & y are nothing but the two sides of the right angled triangle. So, doesn't x = y itself imply that the triangle is iso. rt angled triangle?In that case we already get to know that the angles are equal. Please let me know if I am missing something/understanding is incorrect.

TIA.

TIA.

Bunuel · Answer

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, NOT x and y. So, x = y does NOT mean that we have a right isosceles triangle.

RenanBragion · Answer

I think this is a  high-quality  question and I agree with explanation.

jkrishh7 · Answer

*******************************************************************************
Forget about x-y plane as of now. Just draw any triangle with coordinates of vertices as (x, 1) (5, 1) and (5, y)

Now considering we know nothing about the triangle type but we know  x \lt 5  and  y \gt 1 , we assume that we will use Hero's Formula (google if you do not know). We must know all the sides of the triangle to use Hero's formula.

Statement 1
x=y so the coordinates become (x, 1) (5, 1) and (5, x) or (y, 1) (5, 1) and (5, y). Still, we cannot calculate distance between coordinates ie length of each side. Clearly insufficient.

Statement 2 angle at the vertex (x,1)=angle at vertex (5,y)
=> (x-5)^2+(1-1)^2=(5-5)^2+(y-1)^2
or (x-5)^2=(y-1)^2
or (x+y-6) (x-y-4)=0 still we cannot find x and y so insufficeinet

Combining Statement 1 and 2

(2x-6)(-4)=0
2x=6
x=3 or x=y=3.
Calculate the length of three sides and apply the hero's formulae

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

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