If points A and B are on the y-axis in the figure, what is the area of

Question

Tough and Tricky questions: Coordinate Geometry.

If points A and B are on the y-axis in the figure, what is the area of equilateral triangle ABC ?

(1) The coordinates of point B are (0, 5√3).

(2) The coordinates of point C are (6, 3√3).

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Official Answer

B

navkaran · Accepted Answer

Well if we only have coordinates of B, then point C could be anywhere on the quadrant and Point A anywhere on Y axis. So 1) not sufficient

If we have coordinates of C, and it is given that A and B are on y axis, then we can conclude that the length of the altitude/height from point C to the base would be the X coordinate of C  (3\sqrt{3}) 
So length of height = \sqrt{3} /2 * side a.
Thus side a would be 6.
From the side length we can find out area of equilateral triangle from formula  \sqrt{3} /4 * a^2
Thus 2) is sufficient.

sytabish · Answer

Hint: Use the concept of perpendicular bisector. :-D

Its a good one

sanket1991 · Answer

Consider Statement A : we have coordinates of B but no info is given on C or A, hence not sufficient.

Consider Statement B : we have Coordinates of C (6 , 3_/3) 
now x coordinate means perpendicular distance of that point to Y axis which is nothing but height of Triangle
so we know height is 6
we can get side from height and so area.

Temurkhon · Answer

Rephrased question what the lenght of side of triangleis is, because in equilateral triangle A=a^2*(√3/4), where  a  is a side

St.1. The coordinates of point B are (0, 5√3). No information about any side. INSUFFICIENT

St.2. The coordinates of point C are (6, 3√3). No direct information about side, but we know that altitude and median=6. So, half of the side is 6/√3. SUFFICIENT

B

bullseye800 · Answer

Area of equilateral triangle can be arrived if we have length of one side or one height or...due to its symmetrical properties

Stat1: doesnt give us any unique value (neither side/height/...)
=> no sufficient
Stat2: Gives height straight away...i.e using the (x coordinate of the point C) => we can arrive at area of traingle =>stat 2 is sufficient=>
Ans is B

Psiva00734 · Answer

from statatement 1 , we coordinate points of B(0,5root3).
It is insufficient to calcualte area of equilateraltriangle.

statement 2:

co ordintates of point C (6,3root3).

draw altitude from point C on A and B line segment.

Both points A and B are on yaxis. then x=0. then, drawn altitude from C to AB is (0,3root3)

then height is 6.
we know altitude or height of equilateral triangle is root3/2*a. then we can find side and area also.
Option B is correct

nzk5053 · Answer

Psiva00734

hello, Thank you for the explanation. I understand that the height of this trianble is 6, However, I don't understand how to get the distance from A to AB/2 (this mean half way to point B). To get this wouldn't we have to know the y - distance from point (0,0) to A?

Psiva00734 · Answer

hi nzk503,
In equilateral traingle
three sides are equal
we know altitude is root3/2a

here altitude is 6.= root3/2a=6
then a= 4root3.

In triangle , the line joining drawn from midpoint of a side pass through opposite vertex is called median and it divide trinagle into two equal areas.
Hope i clarify your doubt

sgrover18 · Answer

I think there is a small mistake in your calculation. Just wanted to correct so that others don't get confused.

X coordinate of the the point C is 6 and not 3√3

equating this to 6, we get side a is equal to 4√3. And using this we can get the area.

P.S : There is no need of to calculate while solving this question till the time you know you have enough information to get the length of one side.

Thanks and Regards,
Shradha

Archit3110 · Answer

#1
The coordinates of point B are (0, 5√3).
clearly insufficient
as we dont know the distance from origin
#2
The coordinates of point C are (6, 3√3).
the height of ∆ equilateral is 6 ; 
6=√3/2 * a ; a= 12/√3 ;
sufficient to find area ; 
IMO B

Crytiocanalyst · Answer

1) The coordinates of point B are (0, 5√3).
clearly insufficient

(2) The coordinates of point C are (6, 3√3).
it is given that A and B are on y axis, 
height from point C to the base would be the X coordinate of C (3√3)
So length of height =√3/2 * a.
Thus side a would be 6.
From the side length we can find out area of equilateral triangle from formula 3√3/4 * a^2
Thus sufficient 
Therefore IMO B

imaityaroy · Answer

Is this a focus edition question ?

If points A and B are on the y-axis in the figure, what is the area of

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