What are the coordinates of point B in the figure above ?

IMO: D

Given ∠CAB = 90 . Thus Line AB is parallel to x-axis and the y cordinate will always be "4"

St 1: The area of ∆ABC = 30

Area = \(\frac{1}{2}\)breadth * height
30 = \(\frac{1}{2}\) AC * AB
30 = \(\frac{1}{2}\) 5* AB
AB = 12
Thus Coordinates of B( 9,4)
Suff


St 2: Length of CB = 13

C(-3,-1) B(x,4)
Using Distance formula we can calculate the value of x
Which will be 9
Suff

B(x,4); we have to find out x.
1. 1/2* 5*x=30 ; so we can find out value of x. sufficient.
2. distance between (-3,-1) and (x,4) is 13 so we will be able to find out value of x. sufficient.

Hence answer is D

AC and AB are perpendicular, and we can see slope of AC = (4-(-1))/(-3-(-3)) = infinity
so slope of AB must be 0. Or we can say that since AC is parallel to y-axis, AB must be parallel to x-axis hence its slope must be 0.

This means the y-coordinate of point B must be same as y-coordinate of point A. Thus we can say that point B is (x,4). Our task is to find the value of x and the question is solved. If distance AB is known then also point B can be easily calculated.
We also know that distance AC = 5 (apply distance formula or its just plain visible).

Statement 1. Area of a right angled triangle = 1/2 * product of two perpendicular sides.
Ac is known so AB can be calculated. Thus x can be found. Sufficient.

Statement 2. Length of CB=13. This is a right angle triangle, so using Pythagoras theorem, AB can be calculated.
Thus x can be found. Sufficient.

Hence answer is D

Question: Co-ordinates of point B ?

Observation 1: that the given triangle ABC is a right triangle with one of the legs AC of length 5 units [Distance between (-3,4) and (-3,-1)]
Observation 2: Line AB is parallel to X-Axis so all we need is X-coordinate of point B or length of length of AC to find length of AB using pythagorus theorem

Statement 1: Area of ABC = 30
i.e. (1/2)*AB*AC = 30
i.e. (1/2)*AB*5 = 30
i.e. AB = 12
SUFFICIENT

Statement 2: BC=13
Using pythagorus theorem we find that AB = 12
SUFFICIENT

Answer: Option D
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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 video below.

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

Answer: D

RELATED VIDEO

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Okay so maybe this is just a pattern that im noticing but Sub 600 level questions such as these rarely need you to solve for the final value. So in this question the base of the triangle is given . And the question can be reframed as can the height of the triangle be found given base b .
1. Area is given . Since A = 1/2 b * h it is sufficient to find h.
2. Hypotenuse is given and base is given hence the available information is sufficient to find the height.

Answer is D

No calculations needed .