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# What are the coordinates of point B ?

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Re: What are the coordinates of point B ? [#permalink]
pacifist85 wrote:
What are the coordinates of point B ?

(1) The area of triangle ABC = 30
(2) Length(CB) = 13

hi pacifist,
you dont require to solve anything...
here you have a right angle triangle on x-y axis, where coord of two axis are given...
third vertex coord are reqd..
1) area would give us length of perpendicular.. and hence coord can be found..suff
2) length of hyp given.. vertex can be again found.. suff
ans D
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Re: What are the coordinates of point B ? [#permalink]
Hi

can some one help me how each STATEMENT for the above problem is sufficient ?

with the area and the distance formula . you can calculate the the length of AB ..but how do get the co ordinates or the point B?

and also how is the statement B alone also sufficient ?

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Re: What are the coordinates of point B ? [#permalink]
viper1991 wrote:
Hi

can some one help me how each STATEMENT for the above problem is sufficient ?

with the area and the distance formula . you can calculate the the length of AB ..but how do get the co ordinates or the point B?

and also how is the statement B alone also sufficient ?

Hi viper..

firstly look at the figure and see what it tells us...
1) line AC is parallel to Y axis as the coord of both A and C have same value for X axis..
2) this means AB will be parallel to X axis and A and B will have same Y coord.. or B will be (x,4)

lets look at the statement..
1) it gives us area of triangle 30= base * perpendicular/2... AB=60/5=12 this will give us x as -3 + 12= 9.... sufficient
2)This gives us length of hypotenuse BC=13.. AC is 5 so AB is 12 again as in (1), x coord can be found.. sufficient
ans D
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Re: What are the coordinates of point B ? [#permalink]
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Bunuel, can you please explain this question?
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Re: What are the coordinates of point B ? [#permalink]
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Ergenekon wrote:
Bunuel, can you please explain this question?

Dear Ergenekon

Our first step here would be to observe that since the x- coordinate of both points A and C are the same, this means line AC is parallel to the y - axis.

And from the given diagram, it's clear than line AB is perpendicular to AC. What does this mean? That AB is parallel to x-axis. So, the y-coordinate of point B will be the same as the y-coordinate of point A.

This means, we can write point B as (x, 4)

So, we have only one unknown here: x. Let's see if St. 1 and 2 are sufficient to get us a unique value of x.

St. 1 says
The area of triangle ABC = 30

This means, $$\frac{1}{2}*AB*AC = 30$$

AC = 5 (because point A is 4 units above x-axis and point C is 1 units below x-axis. So, total length of AC = 5)

Similarly AB = 3 +x

So, we can write: $$\frac{1}{2}*(3+x)*5 = 30$$

Linear equation with one unknown. Sufficient to find a unique value of x.

St. 2 says
Length(CB) = 13

In right triangle BAC, using Pythagoras Theorem,

$$AB^2 + AC^2 = BC^2$$
$$AB^2 +5 ^2 = 13^2$$
$$AB^2 = 12^2$$

Since the length of a side cannot be negative, AB cannot be -12. So, the only value of AB that satisfies the above equation is AB = 12

This means, 3 + x = 12

From here, we get x. St. 2 is sufficient as well.

Please let me know if something remains unclear.

Best Regards

Japinder
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Re: What are the coordinates of point B ? [#permalink]
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pacifist85 wrote:
What are the coordinates of point B ?

(1) The area of triangle ABC = 30
(2) Length(CB) = 13

Statement 1

Because this is a right triangle we only need the base and height in order to calculate the area; we know the height of the triangle so we must know the base

base x height/ 2 = 30
base x height =60
base x (5) =60
base= 60/5

Furthermore, because know the x coordinate (-3,4) we simply add 12 to the x value in order to find b-
l-3l + b =12

Statement 2

The diagram shows us that this is a right triangle- well, there is a Pythagorean Triplet 5 12 13- and because we know the height of the triangle we can easily deduce this is a 5 12 13 Pythagorean Right Triangle

Sufficient

Thus
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Re: What are the coordinates of point B ? [#permalink]
GMATPrepNow wrote:
pacifist85 wrote:
What are the coordinates of point B ?

(1) The area of triangle ABC = 30
(2) Length(CB) = 13

IMPORTANT: For geometry Data Sufficiency questions, we are typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the following video: https://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1103

Target question: What are the coordinates of point B ?

NOTE: points A and C are LOCKED in their positions. Since ∠CAB = 90º, we know that point B is SOMEWHERE along the line y = 4. So, some of the MANY possible cases are as follows:

Notice that, for EACH different position of point B, ∆ABC has a different area and side CB has a different length.

Okay, onto the statements...
Statement 1: The area of ∆ABC = 30
As I mentioned above, for EACH different position of point B, ∆ABC has a different area.
So, knowing that the area is 30, LOCKS point B into ONE AND ONLY ONE location.
In other words, statement 1 LOCKS IN the shape/dimensions of ∆ABC, which means there must be only one location for point B.
As such, statement 1 is SUFFICIENT

Statement 2: Length of CB = 13
As I mentioned above, for EACH different position of point B, side CB has a different length.
So, knowing that side CB has length 13 LOCKS point B into ONE AND ONLY ONE location.
In other words, statement 2 LOCKS IN the shape/dimensions of ∆ABC, which means there must be only one location for point B.
As such, statement 2 is SUFFICIENT

Hi, I have a question. What if point B is on the left side of AC, then B will have a different location, leading to the point that there will be two locations of B in both statement 1 and 2? Correct me if I am wrong.
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What are the coordinates of point B ? [#permalink]
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churuand wrote:
Hi, I have a question. What if point B is on the left side of AC, then B will have a different location, leading to the point that there will be two locations of B in both statement 1 and 2? Correct me if I am wrong.

Great question!
This is one of those cases in which we can assume that point P lies to the right.
For more on assumptions we can make on GMAT geometry questions, watch the following video:

Cheers,
Brent
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Re: What are the coordinates of point B ? [#permalink]
EgmatQuantExpert wrote:
Ergenekon wrote:
Bunuel, can you please explain this question?

Dear Ergenekon

Our first step here would be to observe that since the x- coordinate of both points A and C are the same, this means line AC is parallel to the y - axis.

And from the given diagram, it's clear than line AB is perpendicular to AC. What does this mean? That AB is parallel to x-axis. So, the y-coordinate of point B will be the same as the y-coordinate of point A.

This means, we can write point B as (x, 4)

So, we have only one unknown here: x. Let's see if St. 1 and 2 are sufficient to get us a unique value of x.

St. 1 says
The area of triangle ABC = 30

This means, $$\frac{1}{2}*AB*AC = 30$$

AC = 5 (because point A is 4 units above x-axis and point C is 1 units below x-axis. So, total length of AC = 5)

Similarly AB = 3 +x

So, we can write: $$\frac{1}{2}*(3+x)*5 = 30$$

Linear equation with one unknown. Sufficient to find a unique value of x.

St. 2 says
Length(CB) = 13

In right triangle BAC, using Pythagoras Theorem,

$$AB^2 + AC^2 = BC^2$$
$$AB^2 +5 ^2 = 13^2$$
$$AB^2 = 12^2$$

Since the length of a side cannot be negative, AB cannot be -12. So, the only value of AB that satisfies the above equation is AB = 12

This means, 3 + x = 12

From here, we get x. St. 2 is sufficient as well.

Please let me know if something remains unclear.

Best Regards

Japinder

Great explanation EgmatQuantExpert, in St 1, not quite sure why is 3+x and not -3+x since x coordinate of point A is -3? Thanks Japinder
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Re: What are the coordinates of point B ? [#permalink]
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