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Geometry - Using Slope Concept Effectively

The concept of slope is extremely important so it is not sufficient to know how you calculate it using

(y2 - y1)/(x2 - x1). 

 

In simple terms, the slope specifies the units by which the y co-ordinate changes and the direction in which it changes with 1 unit increase in x co-ordinate. If the slope, m, is positive, the y co-ordinate changes in the same direction as the x co-ordinate. If m is negative, the y co-ordinate changes in the opposite direction.  

 

For example, if the slope of a line is 2, it means this:

 

If the x co-ordinate increases by 1 unit, the y co-ordinate will increase by 2 units. So if the point (3, 5) lies on a line with slope 2, the point (4, 7) will also lie on it. Here, when the x co-ordinate increases from 3 to 4, the y co-ordinate will increase from 5 to 7 (increase of 2 units). Similarly,  the point (2, 3) will also lie on the same line. If the x co-ordinate decreases by 1 unit (from 3 to 2), the y co-ordinate will decrease by 2 units (from 5 to 3). Since the slope is positive, the direction of change of x co-ordinate will be the same as the direction of change of y co-ordinate. 

 

Similarly, if the slope of a line is -2 and the point (3, 5) lies on it, when the x co-ordinate increases by 1 unit, the y co-ordinate DECREASES by 2 units. So the point (4, 3) will also lie on the line. Similarly, if the x co-ordinate decreases by 1 unit, the y co-ordinate will increase by 2 units. So the point (2, 7) will also lie on the line. 

 

This understanding of the slope concept can be very helpful as we will see in this question. 

 

Line L and line K have slopes -2 and 1/2 respectively. If line L and line K intersect at (6,8), what is the distance between the x-intercept of line L and the y-intercept of line K? 

 

(A) 5

(B) 10

(C) 5*Sqrt(5)

(D) 15

(E) 10*Sqrt(5)

 

Solution: 

 

Traditionally, one would solve this question like this:

 

Method 1: Slope of line formula

 

Equation of a line with slope m and constant c is given as y = mx + c.

 

The equations of lines L and K would be

y = (-2)x + a

and

y = (1/2)x + b

 

As both these lines pass through (6,8), substitute x=6 and y=8 to get the values of a and b.

 

8 = (-2)*6 + a

a = 20

 

8 = (1/2)*6 + b

b = 5

 

Thus equations of the 2 lines become

 

y = (-2)x + 20

and

y = (1/2)x + 5

 

x intercept of a line is given by the point where y = 0. So x intercept of line K is given by

0 = (-2)x + 20

x = 10

So line L intersects the x axis at the point (10, 0)

 

y intercept of a lien is given by the point where x = 0. So y intercept of line L is given by

y = (1/2)*0 + 5

y = 5

So line K intersects the y axis at the point (0, 5)

 

Distance between these two points is given by Sqrt ((10 - 0)^2 + (0 - 5)^2) = 5*Sqrt(5)

 

Answer (C)

 

Method 2: Using Slope Concept

 

Now notice how we will solve this question using the concept we discussed above:

 

For line L: 

Slope = -2 means that for every 1 unit increase in x co-ordinate, y co-ordinate decreases by 2. Line L has slope -2 and passes through (6, 8). It's x intercept will have y = 0 i.e. a decrease of 8 so x will increase by 4 to give 6 + 4 = 10. So x intercept is at (10, 0).

 

Line K has slope 1/2 and passes through (6, 8). It's y intercept will have x = 0 i.e. a decrease of 6 in x co-ordinate. This means y will decrease by 1/2 of that i.e. by 3 and will become 8 - 3 = 5. So y intercept is at point (0, 5)

 

Distance between the two points can be found using pythagorean theorem as Sqrt (10^2 + 5^2) = 5*Sqrt(5)

or Distance between two points formula Sqrt ((10 - 0)^2 + (0 - 5)^2) = 5*Sqrt(5)

 

Answer (C)

 

Using the slope concept makes solving this question much less tedious and we save a lot of time. That is the advantage of holistic approaches over the more traditional approaches. 

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