A, B, and C are three points on the line shown in the figure. In the

In this question, the slope concept can be utilized to establish an equation, since the three points A, B and C lie on the same line.
This means, Slope of CA = Slope of BC

If the co-ordinates of two points are (\(x_1\), \(y_1\)) and (\(x_2\), \(y_2\)), then the slope of the line connecting these points is given by \(\frac{(y_2 – y_1)}{(x_2 – x_1)}\).

Let the co-ordinates of point A be (25, a) and the co-ordinates of point B be (35, b). Then,

Slope of CA = \(\frac{(t-a)}{(30 – 25)}\) = \(\frac{(t-a)}{5}\) and

Slope of BC = \(\frac{(b-t)}{(35 – 30)}\) = \(\frac{(b-t)}{5}\).

Since slope of CA = slope of BC, we can say (t-a) = (b-t). This means,

t = \(\frac{(a+b)}{2}\).

To find out if t is equal to 55, we need the values of both a and b. Clearly, these are given in the individual statements.

Therefore, neither statement I alone nor statement II alone is sufficient to answer the question. Answer options A, B and D can be eliminated.

We see that, when we combine the statements, we get the value of a = 50 and b = 60. Plugging in these values in the equation for t, we see that t = 55.
We can answer the question with a definite Yes. Combination of statements is sufficient.

The correct answer option is C.

In a question like this, analyzing the question data is important, especially when such a clear diagram is given which gives you lot of data. Assuming the co-ordinates for A and B is equally important because it gives you a chance to develop an equation.

Analysing the question stem will give you an equation/expression/inequality, which will tell you what is the data that you have to look for in the statements. This will take almost the same time as trying out the individual statements, but will help you build a habit of analyzing the question stems, which will in turn help you in the tougher questions.

Hope this helps
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If we know only one point on the line, that line can rotate in any direction in the plane. We need atleast 2 points to fix a line in the plane or to define the equation of the line.
Hence statement 1 and 2 are insufficient, as they give co-ordinates of only one point on the line. By combining 2 statements, we get 2 distinct points on the line. now we can find any property of that line or any point on that line.

C

Given: A, B, and C are three points on the line shown in the figure.
Asked: In the figure, is t equal to 55?

(1) A = (25, 50)
Since only x = 30 is given, y=t may take multiple values and straight line may have any slope
NOT SUFFICIENT

(2) B = (35, 60)
Since only x = 30 is given, y=t may take multiple values and straight line may have any slope
NOT SUFFICIENT

Combining (1) & (2)
(1) A = (25, 50)
(2) B = (35, 60)
Since (30,t) and lies on line joining A = (25, 50) & B = (35, 60)
t=55 since (30,55) is mid points of line segment joining A = (25, 50) & B = (35, 60)
SUFFICIENT

IMO C
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STATEMENT 1 : With cordinates for A alone we can not find the equation for f (x).
Statement 2: same as A
Together, we can tell the slope is 1and having 1 point and the slope of a line we can find the equation of the line and then tell if t= 55.

Answer is C

KarishmaB avigutman can we avoid using any formula and visualize the number line to solve this question?

Hoozan, Does it make sense to you that, for any segment in the xy-coordinate plane between points A and B, the x coordinate of the midpoint is the average of the two x coordinates (of points A and B), and the y coordinate of the midpoint is the average of the two y coordinates (of points A and B)?

I think most people find that quite intuitive in 1-D (on a number line, the midpoint between two tick marks is at the average of those tick marks). I think this problem I just testing our ability to apply that intuition to 2-D.
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You don't even need any visualisation because they have given the relevant figure too. If we know the points A and B, the line is uniquely defined and given another point C on it and its x co-ordinate, we can find the y co-ordinate. Given just one point, the line is not uniquely defined so knowing only A or only B is not enough. So we know that (C) is the answer.

Let's get the value of t too, just for practice.
Remember that slope of a line is change in y when x increases by 1 unit. If you have two points, you can find the slope and that is constant over all parts of the line.
From A to B, y increases by 10 units when x increases by 10 units. So slope is 1.
From A to C, when x increases by 5 units, y will increase by 5 units too. So t will be 5 units more than 50 (the y co-ordinate of A) and hence t will be 55.

Hoozan
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