Last visit was: 11 Sep 2024, 22:59 It is currently 11 Sep 2024, 22:59
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# The coordinates of points A and C are (0, -3) and (3, 3), respectively

SORT BY:
Tags:
Show Tags
Hide Tags
Senior Manager
Joined: 25 Jun 2011
Status:Finally Done. Admitted in Kellogg for 2015 intake
Posts: 395
Own Kudos [?]: 17290 [86]
Given Kudos: 217
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45
GPA: 2.9
WE:Information Technology (Consulting)
MBA Section Director
Joined: 22 Feb 2012
Affiliations: GMAT Club
Posts: 8792
Own Kudos [?]: 10328 [43]
Given Kudos: 4563
Test: Test
Math Expert
Joined: 02 Sep 2009
Posts: 95451
Own Kudos [?]: 657787 [18]
Given Kudos: 87242
General Discussion
Senior Manager
Joined: 25 Jun 2011
Status:Finally Done. Admitted in Kellogg for 2015 intake
Posts: 395
Own Kudos [?]: 17290 [0]
Given Kudos: 217
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45
GPA: 2.9
WE:Information Technology (Consulting)
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
Thanks Bunuel - I didn't get it sorry. How did you arrive at 2 as the co-ordinate for B? Sorry again. Also, I was trying to solve this by using the distance formula for AC.
Math Expert
Joined: 02 Sep 2009
Posts: 95451
Own Kudos [?]: 657787 [0]
Given Kudos: 87242
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
enigma123
Thanks Bunuel - I didn't get it sorry. How did you arrive at 2 as the co-ordinate for B? Sorry again. Also, I was trying to solve this by using the distance formula for AC.

I just put all five points on a plane and saw that the only acceptable answer is C (2, 1). Look at other 4 points (blue) on the diagram and read my explanation: neither of them can divided AC into ratio 2:1.
Math Expert
Joined: 02 Sep 2009
Posts: 95451
Own Kudos [?]: 657787 [2]
Given Kudos: 87242
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
1
Kudos
1
Bookmarks
Bunuel
AccipiterQ
The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

A (1, -$$\sqrt{5}$$)
B (1, -1)
C (2, 1)
D (1.5, 0)
E ($$\sqrt{5}$$,$$\sqrt{5}$$)

could not get this one for the life of me; I calculated what the length of AC was, but after that didn't know how to solve

Point B is on line AC, two-thirds of the way between Point A and Point C. To find the coordinates of point B, it is helpful to imagine that you are a point traveling along line AC.

When you travel all the way from point A to point C, your x-coordinate changes 3 units (from x = 0 to x = 3). Two-thirds of the way there, at point B, your x-coordinate will have changed 2/3 of this amount, i.e. 2 units. The x-coordinate of B is therefore x = 0 + 2 = 2.
When you travel all the way from point A to point C, your y-coordinate changes 6 units (from y = -3 to y = 3). Two-thirds of the way there, at point B, your y-coordinate will have changed 2/3 of this amount, i.e. 4 units. The y-coordinate of B is therefore y = -3 + 4 = 1.

Thus, the coordinates of point B are (2,1).

The correct answer is C.

Merging similar topics. Please refer to the solutions above.

Similar question to practice:
in-the-rectangular-coordinate-system-above-the-line-y-x-144774.html
in-the-xy-coordinate-plane-is-point-r-equidistant-from-143502.html
in-the-rectangular-coordinate-system-the-line-y-x-is-the-132646.html
in-the-rectangular-coordinate-system-the-line-y-x-is-the-88473.html
in-the-rectangular-coordinate-system-above-the-line-y-x-129932.html
the-line-represented-by-the-equation-y-4-2x-is-the-127770.html
points-m-5-2-and-n-5-8-lie-on-the-xy-127803.html
line-segments-ab-and-cd-are-of-equal-length-and-perpendicula-159799.html

Hope it helps.
Manager
Joined: 21 Oct 2013
Posts: 151
Own Kudos [?]: 220 [2]
Given Kudos: 19
Location: Germany
GMAT 1: 660 Q45 V36
GPA: 3.51
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
1
Kudos
1
Bookmarks
y = mx + n

n = -3 (from point A)
3 = 3m -3 (from point C)
m=2
Equation of line AC: y = 2x - 3

Now put in the given answers. None works except from C and D. Both points lie on the line. From the question we know that AB = 2BC. So lets calculate the middle of the line AC. (Xa+Xb)/2 = 0+3 / 2 = 1.5 and (Ya+Yb)/2 = -3+3 / 2 = 0

SO we have M = (1.5;0) which is answer D. Hence answer C is correct.
Senior Manager
Joined: 08 Dec 2015
Posts: 257
Own Kudos [?]: 118 [1]
Given Kudos: 36
GMAT 1: 600 Q44 V27
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
1
Bookmarks
Want a really long way to solve this? No fancy formulas no estimation on paper.

We have from the known coordinates that the big triangle has sides 3, 6, 3BC.

So, $$3^2+6^2=3BC^2$$

We get that $$bc=\sqrt{5}$$

Note that now we have a smaller triangle

Then, using the proportions given, so AB=2BC, we can determine that the h of the small triangle is 2.
We can also calculate the other side since we know the hypotenuse ($$\sqrt{5}$$)

So: $$\sqrt{5}=2^2+x^2$$ X=1

Then we can use the same concept of proportions AB=2BC to determine that$$\sqrt{5}$$ segment corresponds
to 1\3rd of the big triangle. So, the distance from origin on X-axis to the side (base) of the small triangle is 2. And that means that C (2,y). We got our X.

Now get the linear equation of the hypotenuse of the big triangle. Using the given points A and B, we get that slope=6/3 or 2 and y-intercept is -3

So y=2x-3

Now plug-in the X=2 to get that Y=1

Answer C. Logical deduction + Pythagoras Theorem only.
Attachments

triangle 1.jpg [ 21.16 KiB | Viewed 70235 times ]

VP
Joined: 16 Feb 2015
Posts: 1067
Own Kudos [?]: 1088 [3]
Given Kudos: 30
Location: United States
The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
2
Kudos
1
Bookmarks
Bunuel
The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

A. (1, -√5)
B. (1, -1)
C. (2, 1)
D. (1.5, 0)
E. (√5, √5)

Are You Up For the Challenge: 700 Level Questions

Explanantion: It’s a Section Formula problem (internal division case).

B (x’, y’) is such that

x’ = {mb+ na}/(m + n); y’ = {md + nc}/ (m + n).

In the given situation, a = 0, b = 3, c = -3, d = 3, m = 2 and n = 1.

we have x’ = (2 X 3 + 1 X 0)/3 = 2, and y’ = (2 X 3 + 1 X -3)/3 = 1,

the coordinates of point B are (2, 1).

IMO-C
Director
Joined: 21 Feb 2017
Posts: 507
Own Kudos [?]: 1114 [0]
Given Kudos: 1091
Location: India
GMAT 1: 700 Q47 V39
The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
Bunuel
Bunuel
AccipiterQ
The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

A (1, -$$\sqrt{5}$$)
B (1, -1)
C (2, 1)
D (1.5, 0)
E ($$\sqrt{5}$$,$$\sqrt{5}$$)

could not get this one for the life of me; I calculated what the length of AC was, but after that didn't know how to solve

Point B is on line AC, two-thirds of the way between Point A and Point C. To find the coordinates of point B, it is helpful to imagine that you are a point traveling along line AC.

When you travel all the way from point A to point C, your x-coordinate changes 3 units (from x = 0 to x = 3). Two-thirds of the way there, at point B, your x-coordinate will have changed 2/3 of this amount, i.e. 2 units. The x-coordinate of B is therefore x = 0 + 2 = 2.
When you travel all the way from point A to point C, your y-coordinate changes 6 units (from y = -3 to y = 3). Two-thirds of the way there, at point B, your y-coordinate will have changed 2/3 of this amount, i.e. 4 units. The y-coordinate of B is therefore y = -3 + 4 = 1.

Thus, the coordinates of point B are (2,1).

The correct answer is C.

Merging similar topics. Please refer to the solutions above.

Similar question to practice:
https://gmatclub.com/forum/in-the-rectan ... 44774.html
https://gmatclub.com/forum/in-the-xy-coo ... 43502.html
https://gmatclub.com/forum/in-the-rectan ... 32646.html
https://gmatclub.com/forum/in-the-rectan ... 88473.html
https://gmatclub.com/forum/in-the-rectan ... 29932.html
https://gmatclub.com/forum/the-line-repr ... 27770.html
https://gmatclub.com/forum/points-m-5-2- ... 27803.html
https://gmatclub.com/forum/line-segments ... 59799.html

Hope it helps.

AccipiterQ

Found this on Magoosh on reflected points. Might help you

https://magoosh.com/math/coordinate-geo ... ate-plane/
GMAT Instructor
Joined: 30 Jun 2021
Posts: 57
Own Kudos [?]: 21 [3]
Given Kudos: 10
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
3
Kudos
Hey! So first with all coordinate plane problems you want to draw out the given information on a coordinate plane.

You will find in most cases that coordinate plane problems can be solved simply by drawing out a right triangle from the given line segment/hypotenuse, which in this case is line segment AC with A and C having coordinates (0,-3) and (3,3), respectively.

You can do this by drawing out perpendicular lines to the right from A and down from C that will meet at a point, forming a right triangle.

When you do this you can determine the lengths of the triangle legs using the coordinate values or by simply counting the coordinate plane boxes. You see that A is length 3 and B is length 6.

Next you use the Pythagorean theorem to determine the length of hypotenuse C, the line segment created by points A and C given to us by the problem.

$$a^2 + b^2 = c^2$$

$$(3)^2 + (6)^2 = c^2$$

$$9 + 36 = c^2$$

$$C = \sqrt{45}$$

$$\sqrt{45}$$ can be broken down to $$3\sqrt{5}$$ because $$\sqrt{45}$$ = $$\sqrt{9}$$ x $$\sqrt{5}$$ and $$\sqrt{9}$$ can be simplified to 3 leaving you with 3 x $$\sqrt{5}$$ or $$3\sqrt{5}$$.

So now we know that the length of line segment AC is $$3\sqrt{5}$$.

We also know that there is a point B on this line segment that divides the line segment into AB and BC. We are told that AB = 2BC. AB is twice as long as BC.

Since we don't know the value of BC or AB, it is useful here to represent BC with x. This gives us BC = x and AB = 2x. If the entire length of AC = AB + BC, substitute in the x values to get AC = 2x + x = 3x.

Now we know that AC = $$3\sqrt{5}$$ and AC = 3x. Combine the two formulas to solve for x.

$$3\sqrt{5} = 3x$$

$$x = \sqrt{5}$$

That means that $$AB = 2x = 2\sqrt{5}$$.

Why did we go through this whole process? Because now that we have the length of AB, we have the length of the hypotenuse of a smaller right triangle that is inscribed in the larger original right triangle, and we can use the Pythagorean theorem again to give us an idea of the length of the smaller right triangle legs, which will give us the coordinates of point B.

$$c^2 = a^2 + b^2$$

$$(2\sqrt{5})^2 = a^2 + b^2$$

($$2^2$$ x $$\sqrt{5}^2$$) = $$a^2 + b^2$$ = (4 x 5) = 20

Now we know that $$a^2 + b^2 = 20$$ in the smaller triangle, how can this help us determine the coordinates of point B?

We know that the legs of the original larger right triangle are length 3 and 6. That means they are in a 1:2 ratio.

Because this smaller right triangle is inscribed in the larger one, the two triangles must be similar triangles and their legs must be in the same ratio.

What two values are in a 1:2 ratio that, when squared and added to each other are equal to 20?

2 and 4:

$$2^2 + 4^2 = 4 + 16 = 20$$

Therefore, the lengths of the legs of the smaller right triangle are 2 and 4.

Just like you drew the legs of the original larger right triangle 3 values to the right of point A and then 6 values up (or 6 values down from point C), you can draw these smaller legs 2 values to the left of point A and 4 values up. Since point A has coordinates (0, -3), you get (0 + 2, -3 + 4) or (2, 1).

The final coordinates for point B are therefore (2,1), which is answer choice C.

You could also solve by realizing that AB would be a hypotenuse that is $$\frac{2}{3}$$ the length of AC based on the information provided by the question (AB = 2BC) and therefore the length of the smaller legs would be in the same ratio or $$\frac{2}{3}(3) = 2$$ and $$\frac{2}{3}(6) = 4$$ and then use those lengths to determine the coordinates of point B.

You could also visually solve by drawing the line segment and looking at where $$\frac{2}{3 }$$of the line AC going from A to B would be, but I'm sure you know by now that as much as it can sometimes be helpful in a pinch or when you're not sure, going off visuals alone often backfires on the GMAT.
Intern
Joined: 06 Jan 2021
Posts: 24
Own Kudos [?]: 34 [0]
Given Kudos: 31
GMAT 1: 600 Q42 V30
GMAT 2: 720 Q48 V41
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
Bunuel
enigma123
The coordinates of points A and C are (0, -3) and (3, 3), respectively. If point B lies on line AC between points A and C, and if AB = 2BC, which of the following represents the coordinates of point B?

A. (1, -5)
B. (1, -1)
C. (2, 1)
D. (1.5, 0)
E. (5, 5)

How this can be solved guys?

Even though it's possible to set an algebraic equation to get the answer, it would be much easier to draw the line segment AC and you will literally see the answer:
Attachment:
The coordinates.PNG
Since AB is twice the length of BC then the only acceptable choices is B (2, 1). Two points (1, -5) and (5, 5) does not lie on AC at all, (1.-1) is closer to A than to C and (1.5, 0) divides AC in half.

Haven't you put the two points (1, -root5) and (root5, root5) considering them as 5? I would approximate them to a little more than 2; how can we ensure to the (root5, root5) then?

I created the equation otherwise but both b and c are valid that way; clearly not the most reliable way here.
Intern
Joined: 29 Jul 2021
Posts: 21
Own Kudos [?]: 9 [0]
Given Kudos: 37
Location: India
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
Narenn
Refer the first case (i.e. Formula for INTERNAL DIVISION)

In our case $$\frac{m}{n} = \frac{2}{1}$$ AND (x1 y1) = (0,-3) (x2 y2) = (3, 3)

X Coordinate = $$\frac{2(3)+0}{(1+2)}$$ ----------> $$\frac{6}{3}$$ ---------> 2

y Coordinate = $$\frac{2(3)+(-3)}{(1+2)}$$ ----------> $$\frac{3}{3}$$ ---------> 1

Hence B (x y) = 2, 1

Refer my this article for brief understanding of this concept.

Hy narren,Can u please help me out with n.I did not get what n is.

Posted from my mobile device
Director
Joined: 26 Jan 2019
Posts: 508
Own Kudos [?]: 461 [0]
Given Kudos: 118
Location: India
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
A simple solution to this would be to only check the X coordinate.
The X coordinate travels 3 units, from 0 to 3
Since point B divides AC in the ratio is 2:1, we can easily spot that point B would have X coordinate as 2
The only such option is C
Intern
Joined: 10 May 2020
Posts: 11
Own Kudos [?]: [0]
Given Kudos: 69
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
GMATNinja
Can you pls tell another approach for this ques , othe then already mentioned in the forum.

It would be great help.

Intern
Joined: 22 Feb 2018
Posts: 3
Own Kudos [?]: 2 [0]
Given Kudos: 14
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
To Find Coordinates of B (x, y)

Draw a line, DC || EB || X Axis, thus triangles ACD and ABE are similar due to AAA Property.

So, $$AB/AC = AE/AD$$ => $$2/3 = AE/AD$$, As AD = 6, then AE = 4, Thus Coordinate for E will be (0, 1).

So, y=1.

Similarly draw a line || y axis passing through point B, and calculate x = 2.

Intern
Joined: 24 Jan 2023
Posts: 1
Own Kudos [?]: 0 [0]
Given Kudos: 13
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
Two points: (0,-3), (3,3)

ratio of length of AB:BC= 2:1

For x co-ordinate of point B:-
Length of line on x-axis: 3-0= 3.
Thus, length of AB on x-axis= 2/(2+1)= 2
Adding length of AB to co-ordinate of A; x- Co-ordinate of point B= 0+2= 2.

Similarly, to find y co-ordinate of point B:-
Length of AB on y-axis; 3-(-3)= 6; 6/(2+1)=4
And y co-ordinate of point B; -3+4=1

Thus co-ordinates of point B: (2,1)

Posted from my mobile device
Re: The coordinates of points A and C are (0, -3) and (3, 3), respectively [#permalink]
Moderator:
Math Expert
95451 posts