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?
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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: 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.

Answer: C.
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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.
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The questions says that AB = 2BC, not that AB=2AC.
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For the Graph,please look at Bunuel's post.

We can find distance between points A & C using the distance formula : \sqrt{(x2-x1)^2 +(y2-y1)^2}

We get \sqrt{(3-0)^2+(3-(-3))^2} ------> \sqrt{45}

Now at point B ---> AB= 2 BC ------> AB+BC =\sqrt{45} or 3BC =\sqrt{45} ----> BC =\sqrt{5}
So distance from C to B will be \sqrt{5}

Now from answer choice, we can see only option C gives as \sqrt{5} as the distance between Point B and C



Bunuel's reply is more crisp and time saving.
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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.

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.

Let co-ordinate of B = \((x_1, y_1)\)
Therefore, \(x_1 - 0 = 2(3 - x_1)\) ==>\(3(x_1) = 6\) => \(x_1 = 2\)
\(y_1 - (-3) = 2 (3 - y_1)\)==> \(3(y_1) = 6 -3\) ==> \(y_1 = 1\)
Ans.: C

The length of AC = √((x1-x2)^2 )+(y1-y2)^2 = √((0-3)^2 )+(-3-3)^2 = √45 = 3√5
From the question AB = 2BC, this means that AB = 2√5 and BC = √5
By applying the length formula In the multiple choices, the correct answer should give us the right lengths of AB and BC.
Choice C does that:
AB = √((x1-x2)^2 )+(y1-y2)^2 = √((0-2)^2 )+(-3-1)^2 = √20 = 2√5
BC = = √((x1-x2)^2 )+(y1-y2)^2 = √((3-2)^2 )+(3-1)^2 = √5