# Show that the given points (3, 4), (-5, 16), (5, 1) are collinear.

Answer

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Hint: In order to solve this problem we must know that the points are collinear means they are on a single line. Let’s calculate slopes to prove that they are collinear.

The given points are (3, 4), (-5, 16), (5, 1)

Let A(3, 4), B(-5, 16), C(5, 1).

We will first check the slope of the line whose initial point is A and final point is B.

So,

As we know the coordinates A(3, 4), B(-5, 16)

Then the slope of the line joining A and B will be,

$\dfrac{{4 - 16}}{{3 - ( - 5)}} = \dfrac{{ - 12}}{8} = - \dfrac{3}{2}$ (i)

Since the slope of the line joining the point (a,b) and (x,y) is,

\[\dfrac{{b - y}}{{a - x}}\]

Now we will find the slope of the line whose initial point is B and final point is C

Therefore, the slope of the line joining the point B(-5, 16), C(5, 1) is,

$\dfrac{{16 - 1}}{{ - 5 - 5}} = \dfrac{{15}}{{ - 10}} = - \dfrac{3}{2}$ (ii)

Hence we saw in equation (i) and (ii) that the slope is equal, and one of the points is common in finding the slope the first and second time that is point B, it means that they are on a single line.

Hence, it is proved that those points are collinear.

Note: Whenever we face such types of problems the key concept that we need to recall is that some pairs of points will always be collinear. So if all the slopes are equal then the points are collinear. Remember never equate only slope without having a common point in all the slope calculations and say the slope equal, it is not correct, since we can say that they may not be in the same line, maybe the slopes are equal. So there must be a common point in all the calculations of slope to check collinearity of points.

The given points are (3, 4), (-5, 16), (5, 1)

Let A(3, 4), B(-5, 16), C(5, 1).

We will first check the slope of the line whose initial point is A and final point is B.

So,

As we know the coordinates A(3, 4), B(-5, 16)

Then the slope of the line joining A and B will be,

$\dfrac{{4 - 16}}{{3 - ( - 5)}} = \dfrac{{ - 12}}{8} = - \dfrac{3}{2}$ (i)

Since the slope of the line joining the point (a,b) and (x,y) is,

\[\dfrac{{b - y}}{{a - x}}\]

Now we will find the slope of the line whose initial point is B and final point is C

Therefore, the slope of the line joining the point B(-5, 16), C(5, 1) is,

$\dfrac{{16 - 1}}{{ - 5 - 5}} = \dfrac{{15}}{{ - 10}} = - \dfrac{3}{2}$ (ii)

Hence we saw in equation (i) and (ii) that the slope is equal, and one of the points is common in finding the slope the first and second time that is point B, it means that they are on a single line.

Hence, it is proved that those points are collinear.

Note: Whenever we face such types of problems the key concept that we need to recall is that some pairs of points will always be collinear. So if all the slopes are equal then the points are collinear. Remember never equate only slope without having a common point in all the slope calculations and say the slope equal, it is not correct, since we can say that they may not be in the same line, maybe the slopes are equal. So there must be a common point in all the calculations of slope to check collinearity of points.

Last updated date: 25th Sep 2023

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